{"id":3732,"date":"2024-01-30T12:30:16","date_gmt":"2024-01-30T06:30:16","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3732"},"modified":"2024-11-07T16:16:31","modified_gmt":"2024-11-07T10:16:31","slug":"polynomial-and-polynomial-equations","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/polynomial-and-polynomial-equations\/","title":{"rendered":"\u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u0993 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3: \u09ac\u09bf\u09b8\u09cd\u09a4\u09be\u09b0\u09bf\u09a4 (Polynomial and polynomial equations)"},"content":{"rendered":"<h2><\/h2>\n<h2><b><span style=\"color: #339966;\">\u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u0993 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 <span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/academic\/10\/\">\u09b8\u09ae\u09c0\u0995\u09b0\u09a3<\/a><\/span><\/span><\/b><\/h2>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">a_{0}, a_{1}, a_{2}, \\ldots \\ldots \\ldots, a_{n}<\/span> \u09aa\u09cd\u09b0\u09a4\u09cd\u09af\u09c7\u0995\u09c7\u0987 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u0985\u09b0\u09cd\u09a5\u09be\u09ce <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\"> \u09ac\u09b0\u09cd\u099c\u09bf\u09a4 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b8\u0982\u0996\u09cd\u09af\u09be, <\/span><span style=\"font-weight: 400;\">a<\/span><span style=\"font-weight: 400;\">0<\/span><span style=\"font-weight: 400;\">\u22600<\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <\/span><span style=\"font-weight: 400;\">n\u2208N\u222a{0}<\/span><span style=\"font-weight: 400;\">\u00a0 \u09b9\u09b2\u09c7, <span class=\"katex-eq\" data-katex-display=\"false\">P(x)=a_{0} x^{n}+a_{1} x^{n-1}+a_{2} x^{n-2}+\\ldots \\ldots \\ldots+a_{n}<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u0986\u0995\u09be\u09b0\u09c7\u09b0 \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09b0\u09be\u09b6\u09bf\u0995\u09c7 <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 <\/span><span style=\"font-weight: 400;\">n<\/span><span style=\"font-weight: 400;\"> \u09a4\u09ae \u0998\u09be\u09a4\u09c7\u09b0 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b0\u09be\u09b6\u09bf \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0\u09b0 \u09aa\u09a6\u0997\u09c1\u09b2\u09bf\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u0997\u09b0\u09bf\u09b7\u09cd\u09a0 \u0998\u09be\u09a4\u0995\u09c7 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0\u09b0 \u0998\u09be\u09a4 \u09ac\u09be \u09ae\u09be\u09a4\u09cd\u09b0\u09be (degree) \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0\u09a4\u09c7 \u0997\u09b0\u09bf\u09b7\u09cd\u09a0 \u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09af\u09c1\u0995\u09cd\u09a4 \u09aa\u09a6\u099f\u09bf\u0995\u09c7 \u09ae\u09c2\u0996\u09cd\u09af\u09aa\u09a6 \u098f\u09ac\u0982 \u09ac\u09c3\u09b9\u09a4\u09cd\u09a4\u09ae \u0998\u09be\u09a4 \u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09aa\u09a6\u09c7\u09b0 \u09b8\u09b9\u0997\u0995\u09c7 \u09ae\u09c2\u0996\u09cd\u09af \u09b8\u09b9\u0997 (Coefficient) \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964 <\/span><span style=\"font-weight: 400;\">0<\/span><span style=\"font-weight: 400;\"> \u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09af\u09c1\u0995\u09cd\u09a4 \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u099a\u09b2\u0995-\u09ac\u09b0\u09cd\u099c\u09bf\u09a4 \u09aa\u09a6\u099f\u09bf\u0995\u09c7 \u09a7\u09cd\u09b0\u09c1\u09ac\u09aa\u09a6 \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964 <span class=\"katex-eq\" data-katex-display=\"false\">3 x^{4}+5 x^{3}+2 x^{2}+9 x+1, x<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u099a\u09b2\u0995\u09c7\u09b0 \u098f\u0995\u099f\u09bf \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b0\u09be\u09b6\u09bf, \u09af\u09be\u09b0 \u0998\u09be\u09a4 <\/span><span style=\"font-weight: 400;\">4<\/span><span style=\"font-weight: 400;\">,<\/span><span style=\"font-weight: 400;\"> \u09ae\u09c2\u0996\u09cd\u09af\u09aa\u09a6 <span class=\"katex-eq\" data-katex-display=\"false\">3 x^{4}<\/span><\/span><span style=\"font-weight: 400;\">\u00a0, \u09ae\u09c1\u0996\u09cd\u09af \u09b8\u09b9\u0997 <\/span><span style=\"font-weight: 400;\">3<\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 \u09a7\u09cd\u09b0\u09c1\u09ac\u09aa\u09a6 <\/span><\/p>\n<p><span style=\"font-weight: 400;\">1<\/span><span style=\"font-weight: 400;\">.\u00a0\u00a0<\/span><span style=\"font-weight: 400;\">\u09b2\u0995\u09cd\u09b7\u09a3\u09c0\u09af\u09bc \u09af\u09c7, <span class=\"katex-eq\" data-katex-display=\"false\">3 x^{2}+\\frac{5}{x^{3}}+7<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09b0\u09be\u09b6\u09bf\u099f\u09bf \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09a8\u09af\u09bc\u0964 \u0995\u09c7\u09a8\u09a8\u09be, \u09b0\u09be\u09b6\u09bf\u099f\u09bf\u09b0 \u09a6\u09cd\u09ac\u09bf\u09a4\u09c0\u09af\u09bc \u09aa\u09a6\u09c7 <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u0998\u09be\u09a4 \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 <\/span><span style=\"font-weight: 400;\">(<\/span><span style=\"font-weight: 400;\">&#8211;<\/span><span style=\"font-weight: 400;\">3)<\/span><span style=\"font-weight: 400;\">\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">a^{x}, e^{x}, \\log x, \\ln x<\/span> \u00a0<\/span><span style=\"font-weight: 400;\"> \u098f\u09b0\u09be \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09a8\u09af\u09bc\u0964 <\/span><span style=\"font-weight: 400;\">n=0<\/span><span style=\"font-weight: 400;\"> \u09b9\u09b2\u09c7 <\/span><span style=\"font-weight: 400;\">P(x)<\/span><span style=\"font-weight: 400;\"> \u0995\u09c7 <\/span><span style=\"font-weight: 400;\">0<\/span><span style=\"font-weight: 400;\"> \u0998\u09be\u09a4\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\"> \u099a\u09b2\u0995\u09c7\u09b0 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0\u0995\u09c7 \u09b8\u09be\u09a7\u09be\u09b0\u09a3\u09a4 <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u0998\u09be\u09a4\u09c7\u09b0 \u0985\u09a7\u0983\u0995\u09cd\u09b0\u09ae\u09c7 (\u0985\u09b0\u09cd\u09a5\u09be\u09ce, \u09ae\u09c2\u0996\u09cd\u09af\u09aa\u09a6 \u09a5\u09c7\u0995\u09c7 \u09b6\u09c1\u09b0\u09c1 \u0995\u09b0\u09c7 \u0995\u09cd\u09b0\u09ae\u09c7 \u0995\u09cd\u09b0\u09ae\u09c7 \u09a7\u09cd\u09b0\u09c1\u09ac \u09aa\u09a6 \u09aa\u09b0\u09cd\u09af\u09a8\u09cd\u09a4) \u09ac\u09b0\u09cd\u09a3\u09a8\u09be \u0995\u09b0\u09be \u09b9\u09df\u0964 \u098f\u09b0\u09c2\u09aa \u09ac\u09b0\u09cd\u09a3\u09a8\u09be\u0995\u09c7 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0\u099f\u09bf\u09b0 \u0986\u09a6\u09b0\u09cd\u09b6 \u09b0\u09c2\u09aa (Standard form) \u09ac\u09b2\u09be \u09b9\u09df\u0964\u00a0\u00a0<\/span><\/p>\n<p><img loading=\"lazy\" class=\"aligncenter\" src=\"https:\/\/d1whtlypfis84e.cloudfront.net\/guides\/wp-content\/uploads\/2018\/05\/11061140\/TOP-e1526545890562.jpg\" alt=\"\u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u0993 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\" width=\"573\" height=\"232\" \/><\/p>\n<h2><\/h2>\n<h2><span style=\"color: #339966;\"><b>\u09b8\u09ae\u09ae\u09be\u09a4\u09cd\u09b0\u09bf\u0995 \u0993 \u0985\u09b8\u09ae\u09ae\u09be\u09a4\u09cd\u09b0\u09bf\u0995 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 (<\/b><b>Homogeneous and Non-homogeneous polynomials):<\/b><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">\u0995\u09cb\u09a8\u09cb \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0\u09b0 \u09b8\u0995\u09b2 \u09aa\u09a6\u09c7\u09b0 \u0998\u09be\u09a4 \u09b8\u09ae\u09be\u09a8 \u09b9\u09b2\u09c7 \u0990 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0\u0995\u09c7 \u09b8\u09ae\u09ae\u09be\u09a4\u09cd\u09b0\u09bf\u0995 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u098f\u09ac\u0982 \u09b8\u09ae\u09be\u09a8 \u09a8\u09be \u09b9\u09b2\u09c7 \u09a4\u09be\u0995\u09c7 \u0985\u09b8\u09ae\u09ae\u09be\u09a4\u09cd\u09b0\u09bf\u0995 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964<\/span><span style=\"font-weight: 400;\">\u00a0 <span class=\"katex-eq\" data-katex-display=\"false\">\n\na x^{2}+2 h x y+b y^{2}<\/span><\/span><span style=\"font-weight: 400;\">\u098f\u0995\u099f\u09bf<\/span><span style=\"font-weight: 400;\">\u00a0<\/span> <span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\"> \u0993<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">y<\/span><span style=\"font-weight: 400;\"> \u099a\u09b2\u0995\u09c7\u09b0 \u09a6\u09c1\u0987 \u0998\u09be\u09a4\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09b8\u09ae\u09ae\u09be\u09a4\u09cd\u09b0\u09bf\u0995 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">a x^{2}+b x+c<\/span> \u098f\u0995\u099f\u09bf<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\"> \u099a\u09b2\u0995\u09c7\u09b0 \u09a6\u09c1\u0987 \u0998\u09be\u09a4\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u0985\u09b8\u09ae\u09ae\u09be\u09a4\u09cd\u09b0\u09bf\u0995 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u0995\u09c7\u09a8\u09a8\u09be, \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0\u099f\u09bf\u09a4\u09c7 \u09aa\u09cd\u09b0\u09a5\u09ae \u09aa\u09a6\u09c7\u09b0 \u0998\u09be\u09a4 \u09a6\u09c1\u0987, \u09a6\u09cd\u09ac\u09bf\u09a4\u09c0\u09af\u09bc \u09aa\u09a6\u09c7\u09b0 \u0998\u09be\u09a4 \u098f\u0995 \u098f\u09ac\u0982 \u09a4\u09c3\u09a4\u09c0\u09af\u09bc \u09aa\u09a6\u09c7\u09b0 \u0998\u09be\u09a4 \u09b6\u09c2\u09a8\u09cd\u09af\u0964 \u0985\u09b0\u09cd\u09a5\u09be\u09ce, \u09b8\u0995\u09b2 \u09aa\u09a6\u09c7\u09b0 \u0998\u09be\u09a4 \u09b8\u09ae\u09be\u09a8 \u09a8\u09af\u09bc\u0964<\/span><\/p>\n<h2><b><span style=\"color: #339966;\">\u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 (Polynomial<\/span> <span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/www.youtube.com\/watch?v=508PoKGgt1w&amp;list=PL0dr4HGr8HPiSG7NLKPCX8D8z90tpnuvE\" target=\"_blank\" rel=\"noopener\">equation<\/a><\/span><span style=\"color: #339966;\">):<\/span><\/b><\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sum_{i=0}^{n} a_{i} x^{n-i}=a_{0} x^{n}+a_{1} x^{n-1}+a_{2} x^{n-2}+\\cdots+a_{n-1} x+a_{n}=0<\/span> <\/span>\u0986\u0995\u09be\u09b0\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0995\u09c7 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09ac\u09b2\u09c7\u0964 [\u09af\u09c7\u0996\u09be\u09a8\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">a_{0} \\neq 0 \\text { \u098f\u09ac\u0982 } n \\in N<\/span>]<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">\u09aa\u09a6\u09b8\u09ae\u09c2\u09b9\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 \u0995\u09cb\u09a8 \u099a\u09b2\u0995\u09c7\u09b0 \u09b8\u09b0\u09cd\u09ac\u09cb\u099a\u09cd\u099a \u0998\u09be\u09a4 \u09af\u09a4 \u09a5\u09be\u0995\u09c7 \u09a4\u09be\u0995\u09c7 \u09a4\u09a4 \u0998\u09be\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09ac\u09b2\u09c7\u0964 \u09b8\u09b0\u09cd\u09ac\u09cb\u099a\u09cd\u099a \u0998\u09be\u09a4\u0995\u09c7 \u0989\u0995\u09cd\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09ae\u09be\u09a4\u09cd\u09b0\u09be (Degree) \u09ac\u09b2\u09c7\u0964 <\/span><span style=\"font-weight: 400;\">n\u00a0<\/span><span style=\"font-weight: 400;\">\u0998\u09be\u09a4\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7 <\/span><span style=\"font-weight: 400;\">n<\/span><span style=\"font-weight: 400;\"> \u099f\u09bf \u09ae\u09c2\u09b2 \u09a5\u09be\u0995\u09c7\u0964 \u09af\u09c7\u09ae\u09a8: <span class=\"katex-eq\" data-katex-display=\"false\">x^{3}+1=0<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u098f\u0995\u099f\u09bf \u09a4\u09cd\u09b0\u09bf\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09ae\u09be\u09a4\u09cd\u09b0\u09be <\/span><span style=\"font-weight: 400;\">=3<\/span><span style=\"font-weight: 400;\">\u0964 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 \u09aa\u09a6\u09b8\u09ae\u09c2\u09b9\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 \u0995\u09cb\u09a8 \u099a\u09b2\u0995\u09c7\u09b0 \u0998\u09be\u09a4 \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u09b9\u09b2\u09c7 \u09a4\u09be\u0995\u09c7 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09ac\u09b2\u09be \u09af\u09be\u09ac\u09c7 \u09a8\u09be\u0964 \u09af\u09c7\u09ae\u09a8: <span class=\"katex-eq\" data-katex-display=\"false\">3 x^{3}+\\frac{4}{x^{2}}+9=0<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a8\u09af\u09bc\u0964<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">\u098f\u0995\u09be\u09a7\u09bf\u0995 \u099a\u09b2\u0995 \u09b8\u09ae\u09a8\u09cd\u09ac\u09bf\u09a4 \u0995\u09cb\u09a8 \u09aa\u09a6 \u09a5\u09be\u0995\u09b2\u09c7 \u09b8\u09c7 \u09aa\u09a6\u09c7\u09b0 \u0998\u09be\u09a4 \u09b9\u09af\u09bc \u0989\u09ad\u09af\u09bc \u099a\u09b2\u0995\u09c7\u09b0 \u0998\u09be\u09a4\u09c7\u09b0 \u09af\u09cb\u0997\u09ab\u09b2\u0964 \u09af\u09c7\u09ae\u09a8: <\/span><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">x^{2} y^{2}+x^{3}+y^{3}=0<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u098f\u0995\u099f\u09bf \u099a\u09a4\u09c1\u09b0\u09cd\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0964<\/span><\/li>\n<\/ul>\n<h2><span style=\"color: #339966;\"><b>\u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u0989\u09ce\u09aa\u09be\u09a6\u0995 \u0989\u09aa\u09aa\u09be\u09a6\u09cd\u09af (<\/b><b>Factor theorem of polynomial equations):<\/b><\/span><\/h2>\n<p><b>\u09ac\u09b0\u09cd\u09a3\u09a8\u09be (Explanation):<\/b><span style=\"font-weight: 400;\"> \u09af\u09a6\u09bf <\/span><span style=\"font-weight: 400;\">f (x)<\/span><span style=\"font-weight: 400;\"> \u098f\u0995\u099f\u09bf \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b9\u09af\u09bc \u098f\u09ac\u0982 <\/span><span style=\"font-weight: 400;\">f (<\/span><span style=\"font-weight: 400;\">a) <\/span><span style=\"font-weight: 400;\">= 0<\/span><span style=\"font-weight: 400;\"> \u09b9\u09af\u09bc, \u09a4\u09ac\u09c7 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 <\/span><span style=\"font-weight: 400;\">f (x)<\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u098f\u0995\u099f\u09bf \u0989\u09ce\u09aa\u09be\u09a6\u0995 <\/span><span style=\"font-weight: 400;\">x-a<\/span><span style=\"font-weight: 400;\"> \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a3 (Proof):<\/b><span style=\"font-weight: 400;\"> \u09a7\u09b0\u09bf, <\/span><span style=\"font-weight: 400;\">f (x)<\/span><span style=\"font-weight: 400;\"> \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0\u0995\u09c7<\/span><span style=\"font-weight: 400;\">\u00a0x-a<\/span><span style=\"font-weight: 400;\"> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ad\u09be\u0997 \u0995\u09b0\u09b2\u09c7 \u09ad\u09be\u0997\u09ab\u09b2 <\/span><span style=\"font-weight: 400;\">q(x)<\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 \u09ad\u09be\u0997\u09b6\u09c7\u09b7 <\/span><span style=\"font-weight: 400;\">r<\/span><span style=\"font-weight: 400;\"> \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09af\u09bc\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09a4\u09be\u09b9\u09b2\u09c7 \u09b8\u0982\u099c\u09cd\u099e\u09be\u09a8\u09c1\u09b8\u09be\u09b0\u09c7 <\/span><span style=\"font-weight: 400;\">f(<\/span><span style=\"font-weight: 400;\">x) <\/span><span style=\"font-weight: 400;\">= (<\/span><span style=\"font-weight: 400;\">x-a)<\/span><span style=\"font-weight: 400;\"> q (x) +r<\/span><span style=\"font-weight: 400;\"> \u2026\u2026\u2026\u2026\u2026..<strong>(i)<\/strong><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09ad\u09be\u0997\u09b6\u09c7\u09b7 \u0989\u09aa\u09aa\u09be\u09a6\u09cd\u09af \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, <\/span><span style=\"font-weight: 400;\">r = f(a)<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09c7\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 <strong>(i) \u09a8\u0982<\/strong> \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, <\/span><span style=\"font-weight: 400;\">f(x) = (x-a) q (x)+f (a)<\/span><span style=\"font-weight: 400;\"> \u2026\u2026\u2026\u2026\u2026\u2026\u2026.<strong>(ii)<\/strong><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u098f \u09b6\u09b0\u09cd\u09a4\u09c7 <strong>(ii) \u09a8\u0982<\/strong> \u09b9\u09a4\u09c7 \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09af\u09bc <\/span><span style=\"font-weight: 400;\">f<\/span><span style=\"font-weight: 400;\">(<\/span><span style=\"font-weight: 400;\">x)=(x-a)\u00a0q(x)<\/span><span style=\"font-weight: 400;\"> \u09af\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09c7 <\/span><span style=\"font-weight: 400;\">f(x)<\/span><span style=\"font-weight: 400;\"> \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0, <\/span><span style=\"font-weight: 400;\">x-a<\/span><span style=\"font-weight: 400;\"> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09a8\u09bf\u0983\u09b6\u09c7\u09b7\u09c7 \u09ac\u09bf\u09ad\u09be\u099c\u09cd\u09af\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0985\u09a4\u098f\u09ac, \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 <\/span><span style=\"font-weight: 400;\">f(x)<\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 <\/span><span style=\"font-weight: 400;\">x-a<\/span><span style=\"font-weight: 400;\"> \u098f\u0995\u099f\u09bf \u0989\u09ce\u09aa\u09be\u09a6\u0995\u0964<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><b>\u0989\u09a6\u09be\u09b9\u09b0\u09a3 (Example):<\/b><span style=\"font-weight: 400;\"> \u09a7\u09b0\u09bf, <span class=\"katex-eq\" data-katex-display=\"false\">f(x)=x^{4}-2 x^{3}-21 x^{2}+22 x+40<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u0996\u09be\u09a8\u09c7, <span class=\"katex-eq\" data-katex-display=\"false\">f(-1)=1+2-21-22+40=0<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0985\u09b0\u09cd\u09a5\u09be\u09ce, \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 <\/span><span style=\"font-weight: 400;\">f(x)<\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 <\/span><span style=\"font-weight: 400;\">x- (<\/span><span style=\"font-weight: 400;\">-1) <\/span><span style=\"font-weight: 400;\">= x +1<\/span><span style=\"font-weight: 400;\"> \u098f\u0995\u099f\u09bf \u0989\u09ce\u09aa\u09be\u09a6\u0995\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09a4\u09be\u09b9\u09b2\u09c7, <\/span><i><span style=\"font-weight: 400;\"> <span class=\"katex-eq\" data-katex-display=\"false\">x^{4}-2 x^{3}-21 x^{2}+22 x+40=(x+1)\\left(x^{3}-3 x^{2}-18 x+40\\right)<\/span>\u00a0<\/span><\/i><\/p>\n<p>&nbsp;<\/p>\n<h2><span style=\"color: #339966;\"><b>\u09ad\u09be\u0997\u09b6\u09c7\u09b7 \u0989\u09aa\u09aa\u09be\u09a6\u09cd\u09af (<\/b><b>Remainder Theorem):<\/b><span style=\"font-weight: 400;\">\u00a0<\/span><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">\u0995\u09cb\u09a8\u09cb \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 <\/span><span style=\"font-weight: 400;\">f (x)<\/span><span style=\"font-weight: 400;\"> \u0995\u09c7 <\/span><span style=\"font-weight: 400;\">x-\u03b1<\/span><span style=\"font-weight: 400;\"> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ad\u09be\u0997 \u0995\u09b0\u09b2\u09c7, \u09ad\u09be\u0997\u09b6\u09c7\u09b7 <\/span><span style=\"font-weight: 400;\">f (\u03b1) <\/span><span style=\"font-weight: 400;\">\u09b9\u09ac\u09c7\u0964<\/span><\/p>\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a3 (Proof):<\/b><\/p>\n<p><span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 <\/span><span style=\"font-weight: 400;\">f(x)<\/span><span style=\"font-weight: 400;\"> \u0995\u09c7 <\/span><span style=\"font-weight: 400;\">x-\u03b1<\/span><span style=\"font-weight: 400;\"> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ad\u09be\u0997 \u0995\u09b0\u09b2\u09c7, <\/span><span style=\"font-weight: 400;\">f (x )= (x-\u03b1)\u00a0 Q+R\u2026\u2026 (1) <\/span><span style=\"font-weight: 400;\">\u09b9\u09af\u09bc :<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09af\u09c7\u0996\u09be\u09a8\u09c7 \u09ad\u09be\u0997\u09ab\u09b2 <\/span><span style=\"font-weight: 400;\">Q<\/span><span style=\"font-weight: 400;\"> \u09b9\u099a\u09cd\u099b\u09c7 <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 <\/span><span style=\"font-weight: 400;\">n-1<\/span><span style=\"font-weight: 400;\"> \u0998\u09be\u09a4\u09c7\u09b0 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u098f\u09ac\u0982 \u09ad\u09be\u0997\u09b6\u09c7\u09b7 <\/span><span style=\"font-weight: 400;\">R<\/span><span style=\"font-weight: 400;\"> \u09b9\u099a\u09cd\u099b\u09c7 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995\u0964 \u09a6\u09c7\u0996\u09be\u09a4\u09c7 \u09b9\u09ac\u09c7 \u09af\u09c7, <\/span><span style=\"font-weight: 400;\">R = f (\u03b1)<\/span><\/p>\n<p><span style=\"font-weight: 400;\">(1) \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7 <\/span><span style=\"font-weight: 400;\">x=\u03b1<\/span><span style=\"font-weight: 400;\"> \u09ac\u09b8\u09bf\u09af\u09bc\u09c7 \u09aa\u09be\u0987, <\/span><span style=\"font-weight: 400;\">f (\u03b1) = (\u03b1-\u03b1)\u00a0 Q+R<\/span><span style=\"font-weight: 400;\">\u00a0 <\/span><span style=\"font-weight: 400;\">\u2234R = f(\u03b1)<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><b>\u09aa\u09cd\u09b0\u09a4\u09cd\u09af\u09c7\u0995 <\/b><span style=\"font-weight: 400;\">n<\/span><b>-<\/b><b>\u0998\u09be\u09a4\u09c7\u09b0 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/b><span style=\"font-weight: 400;\">f (x)=0<\/span> <b>\u098f\u09b0 \u0995\u09c7\u09ac\u09b2\u09ae\u09be\u09a4\u09cd\u09b0 <\/b><span style=\"font-weight: 400;\">n<\/span> <b>\u09b8\u0982\u0996\u09cd\u09af\u0995 \u09ae\u09c2\u09b2 \u0986\u099b\u09c7\u0964 (<\/b><b>Every polynomial of degree \"n\" has exactly \"n\" roots)<\/b><\/p>\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a3 (Proof):<\/b> <span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <span class=\"katex-eq\" data-katex-display=\"false\">f(x)=a_{0} x^{n}+a_{1} x^{n-1}+a_{2} x^{n-2}+\\cdots+a_{n}<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u098f\u0995\u099f\u09bf <\/span><span style=\"font-weight: 400;\">n<\/span><span style=\"font-weight: 400;\"> \u0998\u09be\u09a4\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0964\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09ac\u09c0\u099c\u0997\u09a3\u09bf\u09a4\u09c0\u09af\u09bc \u09ae\u09cc\u09b2\u09bf\u0995 \u0989\u09aa\u09aa\u09be\u09a6\u09cd\u09af \u0985\u09a8\u09c1\u09b8\u09be\u09b0\u09c7, \u09aa\u09cd\u09b0\u09a4\u09cd\u09af\u09c7\u0995 <\/span><span style=\"font-weight: 400;\">n (n\u22651)<\/span><span style=\"font-weight: 400;\"> \u0998\u09be\u09a4\u09c7\u09b0 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/span><span style=\"font-weight: 400;\">f(x)=0<\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u0995\u09ae\u09aa\u0995\u09cd\u09b7\u09c7 \u098f\u0995\u099f\u09bf \u098f\u09ac\u0982 \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u0985\u09a5\u09ac\u09be \u0985\u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09ae\u09c2\u09b2 \u09ac\u09bf\u09a6\u09cd\u09af\u09ae\u09be\u09a8\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09a7\u09b0\u09bf, \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/span><span style=\"font-weight: 400;\">f (x)<\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2 <span class=\"katex-eq\" data-katex-display=\"false\">a_{1}<\/span> <\/span><span style=\"font-weight: 400;\">\u0964 \u09a4\u09be\u09b9\u09b2\u09c7 \u0989\u09ce\u09aa\u09be\u09a6\u0995 \u0989\u09aa\u09aa\u09be\u09a6\u09cd\u09af \u0985\u09a8\u09c1\u09af\u09be\u09af\u09bc\u09c0 <\/span><span style=\"font-weight: 400;\">f(x)<\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u098f\u0995\u099f\u09bf \u0989\u09ce\u09aa\u09be\u09a6\u0995 <span class=\"katex-eq\" data-katex-display=\"false\">x-a_{1}<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09c1\u09a4\u09b0\u09be\u0982<span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\left(x-\\alpha_{1}\\right) \\varphi_{1}(x) \\ldots \\ldots \\ldots \\ldots \\ldots(i)<\/span>\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09af\u09c7\u0996\u09be\u09a8\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">\\varphi_{1}(x)<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09b9\u09b2\u09cb <\/span><span style=\"font-weight: 400;\">n\u00a0 - 1<\/span><span style=\"font-weight: 400;\"> \u0998\u09be\u09a4\u09c7\u09b0 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09af\u09be\u09b0 \u09aa\u09cd\u09b0\u09a5\u09ae \u09aa\u09a6 <span class=\"katex-eq\" data-katex-display=\"false\">=a_{0} x^{n-1}<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0986\u09ac\u09be\u09b0 \u09ac\u09c0\u099c\u0997\u09a3\u09bf\u09a4\u09c0\u09af\u09bc \u09ae\u09cc\u09b2\u09bf\u0995 \u0989\u09aa\u09aa\u09be\u09a6\u09cd\u09af \u0985\u09a8\u09c1\u09b8\u09be\u09b0\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\"><\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u098f\u09b0 \u0995\u09ae\u09aa\u0995\u09cd\u09b7\u09c7 \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2 \u09ac\u09bf\u09a6\u09cd\u09af\u09ae\u09be\u09a8\u0964\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09a7\u09b0\u09bf, \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <span class=\"katex-eq\" data-katex-display=\"false\">\\varphi_{1}(x)=0<\/span><\/span><span style=\"font-weight: 400;\">\u098f\u09b0 \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2 <span class=\"katex-eq\" data-katex-display=\"false\">a_{2}<\/span><\/span><span style=\"font-weight: 400;\">\u0964 \u09a4\u09be\u09b9\u09b2\u09c7 \u0989\u09ce\u09aa\u09be\u09a6\u0995 \u0989\u09aa\u09aa\u09be\u09a6\u09cd\u09af \u0985\u09a8\u09c1\u09af\u09be\u09af\u09bc\u09c0 <span class=\"katex-eq\" data-katex-display=\"false\">\\varphi_{1}(x)<\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u098f\u0995\u099f\u09bf \u0989\u09ce\u09aa\u09be\u09a6\u0995 <span class=\"katex-eq\" data-katex-display=\"false\">x-a_{2}<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09c1\u09a4\u09b0\u09be\u0982 <span class=\"katex-eq\" data-katex-display=\"false\">\\varphi_{1}(x)=\\left(x-\\alpha_{2}\\right) \\varphi_{2}(x) \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots \\text { (ii) }<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09af\u09c7\u0996\u09be\u09a8\u09c7, <span class=\"katex-eq\" data-katex-display=\"false\">\\varphi_{2}(x)<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09b9\u09b2\u09cb <\/span><span style=\"font-weight: 400;\">n-2<\/span><span style=\"font-weight: 400;\"> \u0998\u09be\u09a4\u09c7\u09b0 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09af\u09be\u09b0 \u09aa\u09cd\u09b0\u09a5\u09ae \u09aa\u09a6 <span class=\"katex-eq\" data-katex-display=\"false\">=a_{0} x^{n-2}<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u0996\u09a8 (i) \u09a8\u0982 \u098f\u09ac\u0982 (ii) \u09a8\u0982 \u09b9\u09a4\u09c7 \u09aa\u09be\u0987,<span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\left(x-\\alpha_{1}\\right)\\left(x-\\alpha_{2}\\right) \\varphi_{2}(x)<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u09ad\u09be\u09ac\u09c7 \u0985\u0997\u09cd\u09b0\u09b8\u09b0 \u09b9\u09df\u09c7 <\/span><span style=\"font-weight: 400;\">n<\/span><span style=\"font-weight: 400;\"> \u09a7\u09be\u09aa\u09c7\u09b0 \u09aa\u09b0 \u09aa\u09be\u0993\u09df\u09be \u09af\u09be\u09ac\u09c7,\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=\\left(x-a_{1}\\right)\\left(x-\\alpha_{2}\\right)\\left(x-\\alpha_{3}\\right) \\ldots \\ldots \\ldots \\varphi_{n}(x) \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots \\text { (iii) }<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09c1\u09a4\u09b0\u09be\u0982 (iii) \u09a8\u0982 \u09b9\u09a4\u09c7 \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09af\u09bc,<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">f(x)=a_{0}\\left(x-\\alpha_{1}\\right)\\left(x-\\alpha_{2}\\right)\\left(x-\\alpha_{3}\\right) \\ldots \\ldots \\ldots\\left(x-\\alpha_{n}\\right) \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots \\text { (iv) }<\/span>\n<p><span style=\"font-weight: 400;\">\u098f\u0996\u09a8, <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_{i} \\in\\left\\{\\alpha_{1}, \\alpha_{2}, \\alpha_{3}, \\ldots \\ldots, \\alpha_{n}\\right.<\/span> <\/span><span style=\"font-weight: 400;\">\u09b9\u09b2\u09c7 (iv) \u09a8\u0982 \u09b9\u09a4\u09c7 \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09ac\u09c7, <span class=\"katex-eq\" data-katex-display=\"false\">f\\left(a_{i}\\right)=0<\/span> <\/span><span style=\"font-weight: 400;\">\u00a0\u09af\u09c7\u0996\u09be\u09a8\u09c7 <\/span><span style=\"font-weight: 400;\">i\u00a0=\u00a01,\u00a02,\u00a03,\u2026\u2026,\u00a0n<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0985\u09a4\u098f\u09ac, <\/span><span style=\"font-weight: 400;\">f(x)=0<\/span><span style=\"font-weight: 400;\"> \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 <\/span><span style=\"font-weight: 400;\">n<\/span><span style=\"font-weight: 400;\"> \u09b8\u0982\u0996\u09cd\u09af\u0995 \u09ae\u09c2\u09b2 <span class=\"katex-eq\" data-katex-display=\"false\">a_{1}, a_{2}, a_{3}, \\ldots \\ldots, a_{n}<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09ac\u09bf\u09a6\u09cd\u09af\u09ae\u09be\u09a8\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u0996\u09a8 \u09af\u09a6\u09bf <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha \\notin\\left\\{\\alpha_{1}, \\alpha_{2}, \\alpha_{3}, \\ldots \\ldots, \\alpha_{n}\\right\\}<\/span><\/span><span style=\"font-weight: 400;\"> \u09b9\u09af\u09bc, \u09a4\u09ac\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">f(\\alpha)=a_{0}\\left(\\alpha-\\alpha_{1}\\right)\\left(\\alpha-\\alpha_{2}\\right)\\left(\\alpha-\\alpha_{3}\\right) \\ldots \\ldots \\ldots\\left(\\alpha-\\alpha_{n}\\right) \\neq 0<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/span><span style=\"font-weight: 400;\">\u098f\u09b0 <span class=\"katex-eq\" data-katex-display=\"false\">f(x)=0 \\text { \u098f\u09b0 } \\alpha_{1}, \\alpha_{2}, \\alpha_{3}, \\ldots \\ldots, \\alpha_{n}<\/span> <\/span><span style=\"font-weight: 400;\">\u00a0\u098f <\/span><span style=\"font-weight: 400;\">n\u00a0<\/span><span style=\"font-weight: 400;\">\u09b8\u0982\u0996\u09cd\u09af\u0995 \u09ae\u09c2\u09b2 \u09ac\u09cd\u09af\u09a4\u09c0\u09a4 \u0985\u09a8\u09cd\u09af \u0995\u09cb\u09a8\u09cb \u09ae\u09c2\u09b2 \u09ac\u09bf\u09a6\u09cd\u09af\u09ae\u09be\u09a8 \u09a5\u09be\u0995\u09a4\u09c7 \u09aa\u09be\u09b0\u09c7 \u09a8\u09be\u0964<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><b>\u09ae\u09c2\u09b2\u09a6 \u09b8\u09b9\u0997\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u098f\u0995\u099f\u09bf \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u0985\u09ae\u09c2\u09b2\u09a6 \u09ae\u09c2\u09b2\u0997\u09c1\u09b2\u09bf \u09af\u09c1\u0997\u09b2\u09c7 \u09a5\u09be\u0995\u09c7<\/b> <b>(In a polynomial equation with rational coefficients, irrational roots occur in pairs)<\/b><b>\u00a0<\/b><\/p>\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a3 (Proof):<\/b> <span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <\/span><span style=\"font-weight: 400;\">f (x) = 0<\/span><span style=\"font-weight: 400;\"> \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2\u09a6 \u09b8\u09b9\u0997\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\">x=p+\\sqrt{q}<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u098f\u09b0 \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2, \u09af\u09c7\u0996\u09be\u09a8\u09c7 <\/span><span style=\"font-weight: 400;\">P\u2208Q<\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{q}=Q^{\\prime}<\/span> \u09a4\u09be\u09b9\u09b2\u09c7,\u00a0 <span class=\"katex-eq\" data-katex-display=\"false\">f(p+\\sqrt{q})=0 \\ldots \\ldots \\text { (i) }<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0986\u09ac\u09be\u09b0, \u09af\u09c7\u09b9\u09c7\u09a4\u09c1 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 <\/span><span style=\"font-weight: 400;\">f (x)<\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u09b8\u09b9\u0997\u0997\u09c1\u09b2\u09bf \u09ae\u09c2\u09b2\u09a6\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09c1\u09a4\u09b0\u09be\u0982 <span class=\"katex-eq\" data-katex-display=\"false\">f(p+\\sqrt{q})=A+\\sqrt{B}<\/span><\/span><span style=\"font-weight: 400;\">\u00a0......<strong> (ii)<\/strong> \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\">f(p-\\sqrt{q})=A-\\sqrt{B}<\/span><\/span><span style=\"font-weight: 400;\">\u00a0.......<strong>(iii)<\/strong><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09af\u09c7\u0996\u09be\u09a8\u09c7 <\/span><span style=\"font-weight: 400;\">A \u2208 Q<\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{B}=Q^{\\prime}<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u0996\u09a8 <strong>(i) \u09a8\u0982 \u0993 (ii)<\/strong> <strong>\u09a8\u0982<\/strong> \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbf{0}=A+\\sqrt{B}<\/span><\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09ac\u09be, <\/span><span style=\"font-weight: 400;\">A\u00a0=\u00a00,\u00a0B\u00a0=\u00a00<\/span><span style=\"font-weight: 400;\"> [\u0995\u09be\u09b0\u09a3 \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2\u09a6 \u0993 \u098f\u0995\u099f\u09bf \u0985\u09ae\u09c2\u09b2\u09a6 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u09af\u09cb\u0997\u09ab\u09b2 \u09b6\u09c2\u09a8\u09cd\u09af \u09b9\u09a4\u09c7 \u09aa\u09be\u09b0\u09c7 \u09a8\u09be]<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09a4\u09be\u09b9\u09b2\u09c7 <strong>(iii) \u09a8\u0982<\/strong> \u09b9\u09a4\u09c7 \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09af\u09bc <\/span><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">f(p-\\sqrt{q})=0<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2 <span class=\"katex-eq\" data-katex-display=\"false\">p+\\sqrt{q}<\/span> <\/span><span style=\"font-weight: 400;\">\u09b9\u09b2\u09c7 \u0985\u09aa\u09b0 \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2 <span class=\"katex-eq\" data-katex-display=\"false\">p-\\sqrt{q}<\/span><\/span><span style=\"font-weight: 400;\"> \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09af\u09bc \u098f\u09ac\u0982 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4\u0995\u09cd\u09b0\u09ae\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">p-\\sqrt{q}<\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2 \u09b9\u09b2\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">p+\\sqrt{q}<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u0985\u09aa\u09b0 \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2 \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09ac\u09c7\u0964\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0985\u09a4\u098f\u09ac <\/span><span style=\"font-weight: 400;\">\u09ae\u09c2\u09b2\u09a6 \u09b8\u09b9\u0997\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u098f\u0995\u099f\u09bf \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u0985\u09ae\u09c2\u09b2\u09a6 \u09ae\u09c2\u09b2\u0997\u09c1\u09b2\u09bf \u09af\u09c1\u0997\u09b2\u09c7 \u09a5\u09be\u0995\u09c7\u0964<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><b>\u0989\u09a6\u09be\u09b9\u09b0\u09a3 (Example):<\/b> <span class=\"katex-eq\" data-katex-display=\"false\">x^{3}-6 x^{2}+9 x-2=0<\/span><span style=\"font-weight: 400;\">\u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2\u09a6 \u09b8\u09b9\u0997\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u09b0 \u0985\u09ae\u09c2\u09b2\u09a6 \u09af\u09c1\u0997\u09b2 \u09ae\u09c2\u09b2 <\/span><span style=\"font-weight: 400;\">2+\u221a3<\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <\/span><span style=\"font-weight: 400;\">2-\u221a3<\/span><span style=\"font-weight: 400;\"> \u09ac\u09bf\u09a6\u09cd\u09af\u09ae\u09be\u09a8\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0986\u09ac\u09be\u09b0 <span class=\"katex-eq\" data-katex-display=\"false\">x^{3}-(7+\\sqrt{2}) x^{2}+(12+7 \\sqrt{2}) x-12 \\sqrt{2}=0<\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u0995\u099f\u09bf \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0964 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 <span class=\"katex-eq\" data-katex-display=\"false\">x=\\sqrt{2}<\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 <\/span><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">x=-\\sqrt{2}<\/span><\/span><span style=\"font-weight: 400;\"> \u09ae\u09c2\u09b2 \u09a8\u09af\u09bc\u0964 \u098f \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u099f\u09bf \u09ae\u09c2\u09b2\u09a6 \u09b8\u09b9\u0997\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09a8\u09af\u09bc\u0964 \u09ae\u09c2\u09b2\u09a6 \u09b8\u09b9\u0997\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09b9\u09b2\u09c7, \u098f\u09b0 \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2 <span class=\"katex-eq\" data-katex-display=\"false\">x=-\\sqrt{2}<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09c7\u09a4\u0964<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><b>\u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u09b9\u0997\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u098f\u0995\u099f\u09bf \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u0995\u09be\u09b2\u09cd\u09aa\u09a8\u09bf\u0995 (\u0985\u09ac\u09be\u09b8\u09cd\u09a4\u09ac) \u09ae\u09c2\u09b2\u0997\u09c1\u09b2\u09bf \u0985\u09a8\u09c1\u09ac\u09a8\u09cd\u09a7\u09c0 \u09af\u09c1\u0997\u09b2\u09c7<\/b> <b>\u09a5\u09be\u0995\u09c7 (<\/b><b>In a polynomial equation with real coefficients, imaginary roots occur conjugate pairs)<\/b><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a3 (Proof):<\/b> <span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <\/span><span style=\"font-weight: 400;\">f (x) = 0<\/span><span style=\"font-weight: 400;\"> \u098f\u0995\u099f\u09bf \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u09b9\u0997\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u098f\u09ac\u0982 <\/span><span style=\"font-weight: 400;\">x\u00a0=\u00a0p\u00a0+\u00a0iq<\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2, \u09af\u09c7\u0996\u09be\u09a8\u09c7 <\/span><span style=\"font-weight: 400;\">p,q \u2208 R<\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\">i=\\sqrt{-1}<\/span><\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\"> \u09a4\u09be\u09b9\u09b2\u09c7 <\/span><span style=\"font-weight: 400;\">f (p+ iq) = 0<\/span><span style=\"font-weight: 400;\"> ..........(i)<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0986\u09ac\u09be\u09b0, \u09af\u09c7\u09b9\u09c7\u09a4\u09c1 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 <\/span><span style=\"font-weight: 400;\">f(x)<\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u09b8\u09b9\u0997\u0997\u09c1\u09b2\u09bf \u09ac\u09be\u09b8\u09cd\u09a4\u09ac\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09c1\u09a4\u09b0\u09be\u0982 <\/span><span style=\"font-weight: 400;\">f(p+\u00a0iq)\u00a0=\u00a0A\u00a0+\u00a0iB<\/span><span style=\"font-weight: 400;\">..........(ii) \u098f\u09ac\u0982 <\/span><span style=\"font-weight: 400;\">=\u00a0A-iB<\/span><span style=\"font-weight: 400;\">\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026(iii)<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09af\u09c7\u0996\u09be\u09a8\u09c7 <\/span><span style=\"font-weight: 400;\">A,B \u2208 R\u00a0<\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <\/span><span style=\"font-weight: 400;\">i= <span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{-1}<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u0996\u09a8 (i) \u0993 (ii) \u09a8\u0982 \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, <\/span><span style=\"font-weight: 400;\">0=A+iB<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09ac\u09be, <\/span><span style=\"font-weight: 400;\">A\u00a0=\u00a00,\u00a0B\u00a0=\u00a00<\/span><span style=\"font-weight: 400;\"> [\u0995\u09be\u09b0\u09a3 <\/span><span style=\"font-weight: 400;\">A=0<\/span><span style=\"font-weight: 400;\"> \u0993 <\/span><span style=\"font-weight: 400;\">B=0<\/span><span style=\"font-weight: 400;\"> \u09a8\u09be \u09b9\u09b2\u09c7 <\/span><span style=\"font-weight: 400;\">A+iB\u00a0=\u00a00<\/span><span style=\"font-weight: 400;\"> \u09b9\u09a4\u09c7 \u09aa\u09be\u09b0\u09c7 \u09a8\u09be\u0964]<\/span><\/p>\n<p><span style=\"font-weight: 400;\">A\u00a0=\u00a00<\/span><span style=\"font-weight: 400;\"> \u0993 <\/span><span style=\"font-weight: 400;\">B\u00a0=\u00a00<\/span><span style=\"font-weight: 400;\"> \u09ac\u09b8\u09bf\u09af\u09bc\u09c7 (iii) \u09a8\u0982 \u09b9\u09a4\u09c7 \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09af\u09bc, <\/span><span style=\"font-weight: 400;\">f(<\/span><span style=\"font-weight: 400;\">p-iq) <\/span><span style=\"font-weight: 400;\">= 0 <\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2 <\/span><span style=\"font-weight: 400;\">p+iq<\/span><span style=\"font-weight: 400;\"> \u09b9\u09b2\u09c7 \u0985\u09aa\u09b0 \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2 <\/span><span style=\"font-weight: 400;\">p-iq<\/span><span style=\"font-weight: 400;\"> \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09af\u09bc\u0964\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0986\u09ac\u09be\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u0995\u09cd\u09b0\u09ae\u09c7 \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2 <\/span><span style=\"font-weight: 400;\">p-iq<\/span><span style=\"font-weight: 400;\"> \u09b9\u09b2\u09c7 \u0985\u09aa\u09b0 \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2 <\/span><span style=\"font-weight: 400;\">p\u00a0+\u00a0iq<\/span><span style=\"font-weight: 400;\"> \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09ac\u09c7\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0985\u09a4\u098f\u09ac \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u09b9\u0997\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u098f\u0995\u099f\u09bf \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u0995\u09be\u09b2\u09cd\u09aa\u09a8\u09bf\u0995 (\u0985\u09ac\u09be\u09b8\u09cd\u09a4\u09ac) \u09ae\u09c2\u09b2\u0997\u09c1\u09b2\u09bf \u0985\u09a8\u09c1\u09ac\u09a8\u09cd\u09a7\u09c0 \u09af\u09c1\u0997\u09b2\u09c7 \u09a5\u09be\u0995\u09c7\u0964<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><b>\u0989\u09a6\u09be\u09b9\u09b0\u09a3 (Example):<\/b><span style=\"font-weight: 400;\"> \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u09b9\u0997\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u098f\u0995\u099f\u09bf \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <span class=\"katex-eq\" data-katex-display=\"false\">2 x^{3}-9 x^{2}+14-5=0<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u098f\u09b0 \u0995\u09be\u09b2\u09cd\u09aa\u09a8\u09bf\u0995 \u0985\u09a8\u09c1\u09ac\u09a8\u09cd\u09a7\u09c0 \u09af\u09c1\u0997\u09b2 \u09ae\u09c2\u09b2 <\/span><span style=\"font-weight: 400;\">2 + i<\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <\/span><span style=\"font-weight: 400;\">2 - i<\/span><span style=\"font-weight: 400;\"> \u09ac\u09bf\u09a6\u09cd\u09af\u09ae\u09be\u09a8\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0986\u09ac\u09be\u09b0 <span class=\"katex-eq\" data-katex-display=\"false\">x^{3}+(5-i) x^{2}+(6+5 i) x-6 i=0<\/span> <\/span><span style=\"font-weight: 400;\">\u098f\u0995\u099f\u09bf \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0964 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 <\/span><span style=\"font-weight: 400;\">x=i<\/span><span style=\"font-weight: 400;\"> \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 <\/span><span style=\"font-weight: 400;\">x=-i<\/span><span style=\"font-weight: 400;\"> \u09ae\u09c2\u09b2 \u09a8\u09af\u09bc\u0964 \u098f \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u099f\u09bf \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u09b9\u0997\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09a8\u09af\u09bc\u0964 \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u09b9\u0997\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u098f \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2 <\/span><span style=\"font-weight: 400;\">x = i<\/span><span style=\"font-weight: 400;\"> \u09b9\u09b2\u09c7 \u0985\u09aa\u09b0 \u098f\u0995\u099f\u09bf \u09ae\u09c2\u09b2 <\/span><span style=\"font-weight: 400;\">x = -i<\/span><span style=\"font-weight: 400;\"> \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09c7\u09a4\u0964<\/span><\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<p><em><strong>\u098f\u0987\u099a\u098f\u09b8\u09b8\u09bf \u0993 \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be\u09b0\u09cd\u09a5\u09c0\u09a6\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u0995\u09cb\u09b0\u09cd\u09b8\u09b8\u09ae\u09c2\u09b9\u0983<\/strong><\/em><\/p>\n<ul>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-25-online-batch-2-bangla-english-ict\/\">HSC 25 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a \u09e8.\u09e6 (\u09ac\u09be\u0982\u09b2\u09be, \u0987\u0982\u09b0\u09c7\u099c\u09bf, \u09a4\u09a5\u09cd\u09af \u0993 \u09af\u09cb\u0997\u09be\u09af\u09cb\u0997 \u09aa\u09cd\u09b0\u09af\u09c1\u0995\u09cd\u09a4\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-26-online-batch-bangla-english-ict\/\">HSC 26 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ac\u09be\u0982\u09b2\u09be, \u0987\u0982\u09b0\u09c7\u099c\u09bf, \u09a4\u09a5\u09cd\u09af \u0993 \u09af\u09cb\u0997\u09be\u09af\u09cb\u0997 \u09aa\u09cd\u09b0\u09af\u09c1\u0995\u09cd\u09a4\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-2025-online-batch\/\">HSC 25 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ab\u09bf\u099c\u09bf\u0995\u09cd\u09b8, \u0995\u09c7\u09ae\u09bf\u09b8\u09cd\u099f\u09cd\u09b0\u09bf, \u09ae\u09cd\u09af\u09be\u09a5, \u09ac\u09be\u09df\u09cb\u09b2\u099c\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-2026-online-batch\/\">HSC 26 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ab\u09bf\u099c\u09bf\u0995\u09cd\u09b8, \u0995\u09c7\u09ae\u09bf\u09b8\u09cd\u099f\u09cd\u09b0\u09bf, \u09ae\u09cd\u09af\u09be\u09a5, \u09ac\u09be\u09df\u09cb\u09b2\u099c\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/medical-admission-course\/\">\u09ae\u09c7\u09a1\u09bf\u0995\u09c7\u09b2 \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 - \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/dhaka-university-a-unit-admission-course\/\">\u09a2\u09be\u0995\u09be \u09ad\u09be\u09b0\u09cd\u09b8\u09bf\u099f\u09bf A Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 - \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/dhaka-university-b-unit-admission-course\/\">\u09a2\u09be\u0995\u09be \u09ad\u09be\u09b0\u09cd\u09b8\u09bf\u099f\u09bf B Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 - \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/buet-ques-solve\/\">\u09ac\u09c1\u09df\u09c7\u099f \u0995\u09cb\u09b6\u09cd\u099a\u09c7\u09a8 \u09b8\u09b2\u09ad \u0995\u09cb\u09b0\u09cd\u09b8<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/gst-a-unit-admission-course\/\">\u0997\u09c1\u099a\u09cd\u099b A Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 - \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/gst-b-unit-admission-course\/\">\u0997\u09c1\u099a\u09cd\u099b B Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 - \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<\/ul>\n<hr \/>\n<p>&nbsp;<\/p>\n<p><em><strong>\u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u09b8\u09cd\u0995\u09bf\u09b2 \u09a1\u09c7\u09ad\u09c7\u09b2\u09aa\u09ae\u09c7\u09a8\u09cd\u099f \u0995\u09cb\u09b0\u09cd\u09b8\u09b8\u09ae\u09c2\u09b9\u0983<\/strong><\/em><\/p>\n<ul>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/study-abroad-complete-guideline\/\">\u09ac\u09bf\u09a6\u09c7\u09b6\u09c7 \u0989\u099a\u09cd\u099a\u09b6\u09bf\u0995\u09cd\u09b7\u09be: Study Abroad Complete Guideline<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/student-hacks\/\">Student Hacks<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ielts-course\/\">IELTS Course by Munzereen Shahid<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/english-grammar-course\/\">Complete English Grammar Course<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ms-bundle\/\"> Microsoft Office 3 in 1 Bundle<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ghore-boshe-freelancing\/\">\u0998\u09b0\u09c7 \u09ac\u09b8\u09c7 Freelancing<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/facebook-marketing\/\">Facebook Marketing<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/adobe-4-in-1-bundle\/\">Adobe 4 in 1 Bundle<\/a><\/span><\/li>\n<\/ul>\n<hr \/>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><em>\u09e7\u09e6 \u09ae\u09bf\u09a8\u09bf\u099f \u09b8\u09cd\u0995\u09c1\u09b2\u09c7\u09b0 \u0995\u09cd\u09b2\u09be\u09b8\u0997\u09c1\u09b2\u09cb \u0985\u09a8\u09c1\u09b8\u09b0\u09a3 \u0995\u09b0\u09a4\u09c7 \u09ad\u09bf\u099c\u09bf\u099f: <span style=\"color: #993300;\"><strong><a style=\"color: #993300;\" href=\"https:\/\/10minuteschool.com\/?ref=https%3A%2F%2Fblog.10minuteschool.com%2Fwordpress%2F&amp;post_id=78178&amp;blog_category_id=700\">www.10minuteschool.com<\/a><\/strong><\/span><\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u0993 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09aa\u09cd\u09b0\u09a4\u09cd\u09af\u09c7\u0995\u09c7\u0987 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u0985\u09b0\u09cd\u09a5\u09be\u09ce x \u09ac\u09b0\u09cd\u099c\u09bf\u09a4 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b8\u0982\u0996\u09cd\u09af\u09be, a0\u22600 \u098f\u09ac\u0982 n\u2208N\u222a{0}\u00a0 \u09b9\u09b2\u09c7, \u00a0\u0986\u0995\u09be\u09b0\u09c7\u09b0 \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09b0\u09be\u09b6\u09bf\u0995\u09c7 x \u098f\u09b0 n \u09a4\u09ae \u0998\u09be\u09a4\u09c7\u09b0 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0 \u09b0\u09be\u09b6\u09bf \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0\u09b0 \u09aa\u09a6\u0997\u09c1\u09b2\u09bf\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 x \u098f\u09b0 \u0997\u09b0\u09bf\u09b7\u09cd\u09a0 \u0998\u09be\u09a4\u0995\u09c7 \u09ac\u09b9\u09c1\u09aa\u09a6\u09c0\u09b0 \u0998\u09be\u09a4 \u09ac\u09be \u09ae\u09be\u09a4\u09cd\u09b0\u09be (degree) \u09ac\u09b2\u09be<\/p>\n<p> <a class=\"redmore\" href=\"https:\/\/10minuteschool.com\/content\/polynomial-and-polynomial-equations\/\">Read 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