Answer the questions that follow concerning zeros of a cubic polynomial.

MATHEMATICS · Multi-part · AP · AP Precalculus · Polynomial and Rational Functions · Polynomial and Rational Functions

Problem

Answer the questions that follow concerning zeros of a cubic polynomial. (a) State the Factor Theorem. (b) Explain why, if

a

and

b

are two distinct zeros of a cubic polynomial

h(x)

, then

(x-a)(x-b)

must divide

h(x)

. (c) A cubic polynomial

h(x)

has zeros

2

and

-3

. Give a reasoned argument to show that

h(x)

can be written as

(x-2)(x+3)(x-k)

for some real number

k

. (d) If

h(x)=x^{3}+px^{2}+qx+r

, express

p

in terms of the three zeros and describe, in words, how the value of

k

could be found once

p

is known.