factor_polys

factor_polys - Factoring polynomials The following two...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Factoring polynomials The following two theorems are of use in factoring polynomials. The Remainder Theorem When a linear expression = ( ) d x - x a is divided into a polynomial ( ) P x to give a quotient ( ) Q x and remainder R , so that = ( ) P x + ( ) - x a ( ) Q x R ------- (i), then = R ( ) P a , that is, the remainder when a polynomial ( ) P x is divided by = ( ) d x - x a , is the value of ( ) P x at = x a , namely ( ) P a . The Factor Theorem A linear expression of the form = ( ) d x - x a is a factor of a polynomial ( ) P x exactly when = ( ) P a 0. Example 1 : Factor the polynomial = ( ) P x + - - x 3 5 x 2 2 x 24, and find the zeros of ( ) P x . Solution : Suppose that = ( ) P x ( ) + x p ( ) + x q ( ) + x r , where p , q and r are integers. Then = p q r - 24 and p , q and r are integer factors of the constant term - 24 in ( ) P x . The associated zeros - p , - q and - r are also integer factors of - 24. If ( ) P x has only one linear factor = ( ) A x + x p , with p an integer and a companion quadratic factor = ( ) B x + + x 2 m x n , with no integer zeros, such that = ( ) P x ( ) A x ( ) B x = ( ) + x p ( ) + + x 2 m x n , we must have = p n - 24, so that p , or equivalently - p , is still an integer factor of - 24. It is therefore reasonable to look for a zero of ( ) P x which is an integer factor of the constant term - 24. The possiblities to consider for such an integer zero of ( ) P x are: + 1, + 2, + 3, + 4, + 6, + 8, + 12, + 24. It is probably easier to check whether + 1 and + 2 are zeros by direct substitution rather than by using synthetic division. = ( ) P 1 + - - 1 5 2 24 = - 20 0, = ( ) P - 1 - + + - 1 5 2 24 = - 18 0 , = ( ) P 2 + - - 8 20 4 24 = 0. Since = x 2 is a zero of ( ) P x , it follows by the Factor Theorem that - x 2 is a factor of ( ) P x . We can check this by synthetic division, and also find the quotient ( ) Q x when ( ) P x is divided by - x 2. 2 | 1 5 - 2 - 24 2 14 24 ________________________ 1 7 12 | 0 = + - - x 3 5 x 2 2 x 24 ( ) - x 2 ( ) + + x 2 7 x 12 . Hence, by factoring the residual quadratic + + x 2 7 x 12, we see that = ( ) P x ( ) - x 2 ( ) + x 3 ( ) + x 4 . ____________ The corresponding zeros of ( ) P x are 2, - 3 and - 4. Example 2 : Factor the polynomial = ( ) P x + - + x 3 2 x 2 13 x 10, and find the zeros of ( ) P x . Solution : The integer factors of 10 provide candidates for zeros of ( ) P x , that is, + 1, + 2, + 5, + 10. Straight away = ( ) P 1 + - + 1 2 13 10 = 0, so it follows by the Factor Theorem that - x 1 is a factor of ( ) P x . We can find the quotient when ( ) P x is divided by - x 1 by using synthetic division. 1 | 1 2 - 13 10 1 3 - 10 ________________________ 1 3 - 10 | 0 = + - + x 3 2 x 2 13 x 10 ( ) - x 1 ( ) + - x 2 3 x 10 = ( ) - x 1 ( ) - x 2 ( ) + x 5 __________ The zeros of ( ) P x are 1, 2 and - 5.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Example 3 : Factor the polynomial = ( ) P x + + + + x 4 11 x 3 41 x 2 61 x 30, and find the zeros of ( ) P x . Solution
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 14

factor_polys - Factoring polynomials The following two...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online