“You wasted $150,000 on an education you coulda got for $1.50 in late fees at the public library.”

3.6 Real Zeros of a Polynomial Function


Synthetic Division Review
Let’s review synthetic division:
(2 x 3  7 x 2  7 x  30)  ( x  2)
We can use synthetic division to help us completely factor polynomials.


Factoring
Completely factor the following function: f ( x )  2 x 3  3 x 2  5 x  6 Hint: -1 is one zero of the function
How can we find out what factor to start with when using synthetic division????

Rational Zeros Theorem
Let f be a polynomial function of the form f ( x )  rational zero of f a n x n 
, then a p n  1 x n  1  ...
 a
1 x  a p
0 where each coefficient is an integer. If is a q a
 a





Potential Rational Zeros
List the potential rational zeros of the following function: f ( x )  2 x 3  11 x 2  7 x  6

Remainder Theorem
Let f be a polynomial function. If f ( x ) is divided by x
– c , then the remainder is f ( c ) .
Ex : ( x
3 
2 x
2 
4 x

3 )

( x

2 )


Potential Rational Zeros
List the potential rational zeros of the following function.
Then find the zeros and factor over the reals.
f ( x )  2 x 3  11 x 2  7 x  6


Real Zeros of Polynomials
Find the real zeros of f and factor f over the reals.
f ( x )  2 x 3  5 x 2  4 x  12


Real Zeros of Polynomials
Find the real zeros of f and factor f over the reals.
f ( x )  x 4  6 x 3  x 2  26 x  20

Practice Problem
Find the real zeros of f and factor f over the reals.
f x
 x
6  x
5 
6 x
4 
4 x
2 
4 x

24

3.6 Real Zeros of a Polynomial Function
HW #9: p.231 #11 – 19 Odd
39, 41, 63, 69