PP Section 12.6

# PP Section 12.6 - Conjugate Roots and Descartes Rule of...

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Conjugate Roots and Descartes' Rule of Signs

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Note: It follows that a polynomial f of odd degree with real coefficients has at least one real zero. Conjugate Roots Theorem Let f(x) be a polynomial, all of whose coefficients are real numbers. If the complex number a + bi is a root of the equation f(x) = 0, then a – bi is also a root of the equation.
Example Find a polynomial f of degree 4 whose coefficients are real numbers and has zeros 0, -2 and 1 - 3 i . Graph f to verify the solution. Since 1 - 3i is a zero, so is 1 + 3i. )) 3 1 ( ))( 3 1 ( ))( 2 ( )( 0 ( ) ( i x i x x x x f + - - - - - - = f x x x x i x i x i i ( ) ( )( ( ) ( ) ( )( ) = + - + - - + - + 2 1 3 1 3 1 3 1 3 2 f x x x x x ix x ix i i i ( ) ( )( ) = + - - - + + + - - 2 2 2 2 3 3 1 3 3 9 f x x x x x ( ) ( )( ) = + - + 2 2 2 2 10 f x x x x ( ) = + + 4 2 6 20

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Theorem equation. the of root a also is then 0 equation the of root a is and irrational is If c b a f(x) c b a c - = + Let f(x) be a polynomial in which all coefficients are rational numbers. Let a, b, and c be rational numbers. Hence, irrational roots also happen in conjugate pairs.
roots. its of one as 2 5 2 having polynomial quadratic a Find + Example ( 29 (

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