Download 7.1 Synthetic Substitution

Precalculus
Chapter 7: Polynomial Functions
7.1 Synthetic Division and the Remainder and Factor Theorems
Remainder Theorem: When polynomial P (x ) is divided by x  a , the remainder is P (a ) .
Factor Theorem: A number a is a solution of the polynomial equation f ( x)  0 if and only if
x  a is a factor of
f (x) .
Examples:
1.
g x   6 x 3  x 2  19 x  5 , Find g  32 
Use synthetic substitution to evaluate:
2.
g x   6 x 3  x 2  19 x  5 , Find g  32 
3.
g x   9 x 3  3x 2  10 x  15 , Find g  23 
5.
x
Divide, use synthetic division.
4.
2x
3

 3x 2  2 x  4   x  2 
5

 5x 2  x  3
6.
4x
7. Is
5

 7 x 3  9 x 2  4 x  1  2 x  1
x  5 a factor of x 4  3x 3  7 x 2  60 x  125 ? Justify.
Determine k so that the second polynomial is a factor of the first polynomial.
8.
f x   x 4  6 x 3  12 x 2  k  10 ; x  1
9.
g x   x 5  5x 4  7 x 3  kx 2  4 x  20 ; x  5