# Zeros of a Polynomial Function

__ REVIEW: __To find the x-intercepts of f(x) =
2x

^{2}– 5x – 3 we substitute 0 for y and solve for x to

obtain:

**NOTES:
**The x-intercepts of a polynomial function f(x) are also called “

**real zeros**” or

**“real roots**” of the

function. The x-intercepts of the function are the

__real__solutions of the equation f(x) =0.

In the example above, -1/2 and 3 are real solutions of 0 =
2x^{2} – 5x – 3. They are real zeros or

real roots of f(x) = 2x^{2} – 5x - 3.

When x = 3 is an x-intercept of the function, then x-3 is
a factor of the polynomial 2x^{2} –5x –3.

This means that when 2x^{2} – 5x – 3 is divided by x-3, the remainder will be
zero.

__ Exercise:__ Determine the real zeros of the
function f(x) = x

^{3}– 2x

^{2}– 3x, then sketch the graph.

__ Verifying Real Zeros of a Function Graphically:__1. Examine the graph of the function and estimate the value of the "real
zeros" by

observing the x-intercept(s).

2. Verify that the "Observed zero" (x-intercept) is truly a zero of the given function by

a. plugging the value in for x and checking that a value of 0 is obtained for f(x), OR

b. using Synthetic Division with the observed zero and checking for a remainder of 0

(indicating that a factor of the polynomial is (x – r) where r is the verified root)

**Note: a difference between the possible number of zeros

and the actual number of real zeros indicates the existence of imaginary zeros.

Exercises:

1. Given the graph at the right of f(x) = x^{3} – 3x^{2} + x –
3

Verify the observed zero by plugging the value in for x.

Note: How many REAL zeros are there? ________

How many IMAGINARY zeros are there? ________

2. Given the graph at the right of f(x) = -x^{3} + 3x^{2} + 3x
+ 4

Verify the observed REAL zero using synthetic division:

How many REAL zeros are there? _____

How many IMAGINARY zeros are there? _____

For each of the following functions, graph using your
calculator, determine the real zero(s) by

observation, then verify the real zero(s) using synthetic division. Make a note
of the number of

real zeros, and the number of imaginary zeros for each function.

3. f(x) = 3x^{3} + 11x^{2} – 2x + 8

4. f(x) = x^{3} + 7x^{2} + 7x – 15 (Multiple roots)

5. f(x) = x^{3} + 4x^{2} - 11x + 6 (Repeated root –“touch and turn”)

6. f(x) = 2x^{3} + 5x^{2} + 9 (Missing terms)

7. f(x) = 3x^{3} + 10x^{2} – x – 12 (Non – integer roots)

HW: Use these directions, not the directions in the text.

Page 265 Verify any one of the given zeroes by using synthetic division on
problems

25 – 30. (041)