{"id":2151,"date":"2020-07-30T09:31:59","date_gmt":"2020-07-30T09:31:59","guid":{"rendered":"http:\/\/ibalmaths.com\/?post_type=knowledgebase&p=2151"},"modified":"2020-08-01T10:34:49","modified_gmt":"2020-08-01T10:34:49","slug":"notes-polynomials","status":"publish","type":"knowledgebase","link":"https:\/\/ibalmaths.com\/index.php\/ibdp-math-hl-2\/polynomials\/notes-polynomials\/","title":{"rendered":"Notes – Polynomials"},"content":{"rendered":"

A polynomial function, \r\np<\/mi>x<\/mi><\/mfenced><\/math>\r\n , is an algebraic expression that takes the form\u00a0\u00a0 \r\np<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>a<\/mi>n<\/mi><\/msub>x<\/mi>n<\/mi><\/msup>+<\/mo>a<\/mi>n<\/mi>–<\/mo>1<\/mn><\/mrow><\/msub>x<\/mi>n<\/mi>–<\/mo>1<\/mn><\/mrow><\/msup>+<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>+<\/mo>a<\/mi>2<\/mn><\/msub>x<\/mi>2<\/mn><\/msup>+<\/mo>a<\/mi>1<\/mn><\/msub>x<\/mi>+<\/mo>a<\/mi>o<\/mi><\/msub> <\/mo>,<\/mo> <\/mo>a<\/mi>n<\/mi><\/msub>≠<\/mo>0<\/mn><\/math>\r\n<\/p>\r\n

where the coefficients\u00a0 \r\na<\/mi>n<\/mi><\/msub> <\/mo>,<\/mo> <\/mo>a<\/mi>n<\/mi>–<\/mo>1<\/mn><\/mrow><\/msub> <\/mo>,<\/mo> <\/mo>a<\/mi>n<\/mi>–<\/mo>2<\/mn><\/mrow><\/msub> <\/mo>,<\/mo> <\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo> <\/mo>,<\/mo> <\/mo>a<\/mi>1<\/mn><\/msub> <\/mo>,<\/mo> <\/mo>a<\/mi>0<\/mn><\/msub><\/math>\r\n are real numbers, and the powers \r\nn<\/mi>,<\/mo> <\/mo>n<\/mi>–<\/mo>1<\/mn> <\/mo>,<\/mo> <\/mo>n<\/mi>–<\/mo>2<\/mn> <\/mo>,<\/mo> <\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo><\/math>\r\n are non- negative integers
\r\n
\r\nThe degree of a polynomial is the highest power of\u00a0 \r\nx<\/mi><\/math>\r\n in the expression.<\/p>\r\n

SYNTHETIC DIVISION<\/strong><\/span><\/p>\r\n

Dividing a cubic polynomial\u00a0 \r\np<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>a<\/mi>3<\/mn><\/msub>x<\/mi>3<\/mn><\/msup>+<\/mo>a<\/mi>2<\/mn><\/msub>x<\/mi>2<\/mn><\/msup>+<\/mo>a<\/mi>1<\/mn><\/msub>x<\/mi>+<\/mo>a<\/mi>0<\/mn><\/msub><\/math>\r\n by a linear polynomial\u00a0 \r\nx<\/mi>–<\/mo>k<\/mi><\/math>\r\n .<\/p>\r\n

\"\"<\/p>\r\n

Quotient =\u00a0 \r\na<\/mi>3<\/mn><\/msub>x<\/mi>2<\/mn><\/msup>+<\/mo>b<\/mi>1<\/mn><\/msub>x<\/mi>+<\/mo>b<\/mi>o<\/mi><\/msub><\/math>\r\n \u00a0 , Remainder = R<\/p>\r\n

THE REMAINDER THEOREM<\/strong><\/span><\/p>\r\n

For any polynomial \r\np<\/mi>x<\/mi><\/mfenced><\/math>\r\n , the remainder when divided by\u00a0 \r\nx<\/mi>–<\/mo>α<\/mi><\/mrow><\/mfenced><\/math>\r\n is \r\np<\/mi>α<\/mi><\/mfenced><\/math>\r\n .<\/p>\r\n

Q.\u00a0<\/strong>Find the remainder when\u00a0 \r\n3<\/mn>x<\/mi>4<\/mn><\/msup>+<\/mo>4<\/mn>x<\/mi>2<\/mn><\/msup>–<\/mo>2<\/mn>x<\/mi>+<\/mo>1<\/mn><\/math>\r\n is divided by\u00a0 \r\nx<\/mi>+<\/mo>2<\/mn><\/math>\r\n .<\/p>\r\n

Remainder =\u00a0 \r\np<\/mi>–<\/mo>2<\/mn><\/mrow><\/mfenced>=<\/mo>3<\/mn>–<\/mo>2<\/mn><\/mrow><\/mfenced>4<\/mn><\/msup>+<\/mo>4<\/mn>–<\/mo>2<\/mn><\/mrow><\/mfenced>2<\/mn><\/msup>–<\/mo>2<\/mn>–<\/mo>2<\/mn><\/mrow><\/mfenced>+<\/mo>1<\/mn> <\/mo>=<\/mo> <\/mo>69<\/mn><\/math>\r\n<\/p>\r\n

THE FACTOR THEOREM<\/strong><\/span><\/p>\r\n

\r\nx<\/mi>–<\/mo>α<\/mi><\/mrow><\/mfenced><\/math>\r\n is a factor of\u00a0 \r\np<\/mi>x<\/mi><\/mfenced><\/math>\r\n if and only if\u00a0 \r\np<\/mi>α<\/mi><\/mfenced>=<\/mo>0<\/mn><\/math>\r\n .<\/p>\r\n

Q. <\/strong>Find the value of\u00a0 \r\nk<\/mi><\/math>\r\n if\u00a0 \r\nx<\/mi>–<\/mo>1<\/mn><\/math>\r\n is a factor of\u00a0 \r\nh<\/mi>x<\/mi><\/mfenced>=<\/mo>x<\/mi>3<\/mn><\/msup>–<\/mo>k<\/mi>x<\/mi>2<\/mn><\/msup>+<\/mo>2<\/mn>x<\/mi>–<\/mo>1<\/mn><\/math>\r\n .<\/p>\r\n

Since\u00a0 \r\nx<\/mi>–<\/mo>1<\/mn><\/math>\r\n is a factor of\u00a0 \r\nh<\/mi>x<\/mi><\/mfenced><\/math>\r\n , so\u00a0 \r\nh<\/mi>1<\/mn><\/mfenced>=<\/mo>0<\/mn><\/math>\r\n<\/p>\r\n

\r\n⇒<\/mo>1<\/mn>3<\/mn><\/msup>–<\/mo>k<\/mi>1<\/mn><\/mfenced>2<\/mn><\/msup>+<\/mo>2<\/mn>1<\/mn><\/mfenced>–<\/mo>1<\/mn>=<\/mo>0<\/mn><\/math>\r\n<\/p>\r\n

\r\n⇒<\/mo>k<\/mi>=<\/mo>2<\/mn><\/math>\r\n<\/p>\r\n

Given a polynomial \r\na<\/mi>n<\/mi><\/msub>x<\/mi>n<\/mi><\/msup>+<\/mo>a<\/mi>n<\/mi>–<\/mo>1<\/mn><\/mrow><\/msub>x<\/mi>n<\/mi>–<\/mo>1<\/mn><\/mrow><\/msup>+<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>+<\/mo>a<\/mi>2<\/mn><\/msub>x<\/mi>2<\/mn><\/msup>+<\/mo>a<\/mi>1<\/mn><\/msub>x<\/mi>+<\/mo>a<\/mi>o<\/mi><\/msub> <\/mo>,<\/mo> <\/mo>a<\/mi>n<\/mi><\/msub>≠<\/mo>0<\/mn><\/math>\r\n <\/strong><\/span><\/p>\r\n

has a factor <\/strong> \r\np<\/mi>x<\/mi>–<\/mo>q<\/mi><\/mrow><\/mfenced><\/math>\r\n if and only if\u00a0 \r\np<\/mi><\/math>\r\n is a factor of\u00a0 \r\na<\/mi>n<\/mi><\/msub><\/math>\r\n and\u00a0 \r\nq<\/mi><\/math>\r\n is a factor of\u00a0 \r\na<\/mi>0<\/mn><\/msub><\/math>\r\n .<\/strong><\/span><\/p>\r\n

This result is useful in helping us guess potential factors of a given polynomial.<\/p>\r\n

The polynomial: \r\nh<\/mi>x<\/mi><\/mfenced>=<\/mo>x<\/mi>3<\/mn><\/msup>+<\/mo>3<\/mn>x<\/mi>2<\/mn><\/msup>+<\/mo>6<\/mn>x<\/mi>+<\/mo>8<\/mn><\/math>\r\n<\/p>\r\n

can be factorised if we can find a factor (px \u2013 q) where p is a factor of 1 and q is a factor of 4.
\r\nFactors of 1 are 1 \u00d7 1 and factors of 4 are \u00b11 \u00d7 \u00b14 and \u00b12 \u00d7 \u00b12, so possible factors of \r\nh<\/mi>x<\/mi><\/mfenced><\/math>\r\n \u00a0are\u00a0\r\nx<\/mi>±<\/mo>1<\/mn><\/mrow><\/mfenced> <\/mo>,<\/mo> <\/mo>x<\/mi>±<\/mo>2<\/mn><\/mrow><\/mfenced><\/math>\r\n and \r\nx<\/mi>±<\/mo>4<\/mn><\/mrow><\/mfenced><\/math>\r\n.<\/p>\r\n

\r\nx<\/mi>=<\/mo>–<\/mo>2<\/mn><\/math>\r\n , gives\u00a0 \r\nh<\/mi>–<\/mo>2<\/mn><\/mrow><\/mfenced>=<\/mo>0<\/mn><\/math>\r\n . So, \r\nx<\/mi>+<\/mo>2<\/mn><\/mrow><\/mfenced><\/math>\r\n is one of the factors of\u00a0 \r\nh<\/mi>x<\/mi><\/mfenced><\/math>\r\n .<\/p>\r\n

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