Prentice Hall Algebra 2

Chapter 5 Polynomials and Polynomial Functions > 5-2 Polynomials, Linear Factors, and Zeros

5-2 Polynomials, Linear Factors, and Zeros

Objectives

To analyze the factored form of a polynomial

To write a polynomial function from its zeros

A solve it problem with Anya.
Image Long Description

If P(x) is a polynomial function, the solutions of the related polynomial equation p open x close equals 0 are the zeros of the function.

Essential Understanding Finding the zeros of a polynomial function will help you factor the polynomial, graph the function, and solve the related polynomial equation.

In Chapter 4, you solved a quadratic equation of the form x 2 + b x + c = 0 x squared , plus b x plus c equals 0 by factoring. You wrote it using linear factors in the form ( x − r 1 ) ( x − r 2 ) = 0 . open x minus , r sub 1 , close open x minus , r sub 2 , close equals 0 . Then you applied the Zero-Product Property to find the solutions x equals , r sub 1 and x equals , r sub 2 , . You can solve some polynomial equations a n x n + a n − 1 x n − 1 + ⋯ + a 0 = 0 eh sub n , x to the n , plus . eh sub n minus 1 end sub . x super n minus 1 end super . plus math axis ellipsis plus , eh sub 0 , equals 0 in much the same way.

Problem 1 Writing a Polynomial in Factored Form

What is the factored form of x 3 − 2 x 2 − 15 x ? x cubed , minus , 2 x squared , minus 15 x question mark

x 3 − 2 x 2 − 15 x = x ( x 2 − 2 x − 15 ) Factor out the GCF, x . = x ( x − 5 ) ( x + 3 ) Factor x 2 − 2 x − 15. Check x ( x − 5 ) ( x + 3 ) = x ( x 2 − 2 x − 15 ) Multiply ( x − 5 ) ( x + 3 ) . = x 3 − 2 x 2 − 15 x ✓ Distributive Property table with 4 rows and 3 columns , row1 column 1 , x cubed , minus 2 , x squared , minus 15 x , column 2 equals x . open . x squared , minus 2 x minus 15 . close , column 3 cap factoroutthecap gcap ccap fcomma . x . , row2 column 1 , , column 2 equals x . open , x minus 5 , close . open , x plus 3 , close , column 3 cap factor . x squared , minus 2 x minus 15. , row3 column 1 , cap check , x . open , x minus 5 , close . open , x plus 3 , close , column 2 equals x . open . x squared , minus 2 x minus 15 . close , column 3 cap multiply , . open , x minus 5 , close . open , x plus 3 , close . . , row4 column 1 , , column 2 equals , x cubed , minus 2 , x squared , minus 15 x check mark , column 3 cap distributivecap property , end table

✓ Got It?

What is the factored form of x 3 − x 2 − 12 x ? x cubed , minus , x squared , minus 12 x question mark

End ofPage 288

5-2 Polynomials, Linear Factors, and Zeros

Lesson Vocabulary

Problem 1 Writing a Polynomial in Factored Form

Plan

✓ Got It?

Table of Contents