### Section 2.4: Real Zeros of Polynomial Functions

```April 7, 2015
Polynomial Long Division
Divide
by w – 3.
3
2
3
x

2
x
 4 x  3 by x 2  3x  3 .
Divide
Synthetic Division
Divide x 2  5 x  6 by x – 1.
3
Divide 3x  6 x  2 by x – 2.
Remainder Theorem
If dividing f(x) by x – a, f(a) will determine the remainder.
3
What is the remainder when dividing x  7 x  6 by
x – 4?
Use the Remainder Theorem to evaluate f (x) = 6x3 –
5x2 + 4x – 17 at x = 3.
Factor Theorem
When dividing polynomials, if the remainder is zero,
then the divisor is a factor.
Use the Factor Theorem to determine whether x – 1
is a factor of f (x) = 2x4 + 3x2 – 5x + 7.
Using the Factor Theorem, verify that x + 4 is a
factor of f (x) = 5x4 + 16x3 – 15x2 + 8x + 16.
Finding Exact Irrational Zeros
Find the exact zeros for the function. Identify each zero
as rational or irrational.
Writing Functions
w/ Given Conditions
If the zeros of the function are -3, 7, and -1, and the leading
coefficient is 4, write an equation for the function.
Rational Zeros Theorem
Given a function with constant of p and leading coefficient
of q, all possible rational zeros can be found by
factors of p

factors of q
3
2
Find the possible rational zeros of f ( x)  2x  7 x  8x  6.
Find the possible rational zeros of g ( x)  3x3  5x2  9x  16
Upper and Lower Bounds Test
When dividing polynomials, if the quotient polynomial has
all non-negative coefficients, then the “k” value is an upper
bound. If the quotient polynomial has alternating sign
coefficients, then the “k” value is a lower bound.
Show that all real roots of the
equation f ( x)  x5  5x3 10x2  12x  20 lie between - 4 and 4.
```