Lesson Objectives

- Learn how to find the zeros of a polynomial function

## How to Find the Zeros of a Polynomial Function

Over the course of the last few lessons, we have been discussing various tools and techniques that can be used to find the zeros for a polynomial function. Let's walk through an example of how to find the zeros for a polynomial function.

Example #1: Find all zeros. $$f(x)=x^3 - 3x^2 - 12x + 10$$ First, check to see if you can factor the polynomial as it stands. In this case, we can't factor using grouping. Let's move on and think about a few things.

Example #1: Find all zeros. $$f(x)=x^3 - 3x^2 - 12x + 10$$ First, check to see if you can factor the polynomial as it stands. In this case, we can't factor using grouping. Let's move on and think about a few things.

- From the fundamental theorem of algebra, we know that we have at most 3 distinct solutions
- From the rational roots test, we obtain a list of possible rational roots:
- $$\pm (1, 2, 5, 10)$$

- From Descartes' rule of signs, we obtain a list of possible positive and negative real roots:
- + roots: 2 or 0
- - roots: 1

#### Skills Check:

Example #1

Find all zeros. $$f(x)=4x^3 + x^2 - 4x - 1$$

Please choose the best answer.

A

$$x=\frac{2}{3}, -2$$

B

$$x=\frac{1}{2}, 5$$

C

$$x=0, -1, 3$$

D

$$x=-2, 2$$

E

$$x=-1, 1, -\frac{1}{4}$$

Example #2

Find all zeros. $$f(x)=4x^3 + 12x^2 + x + 3$$

Please choose the best answer.

A

$$x=-1, -3, \frac{1}{3}$$

B

$$x=-5, 3, 2$$

C

$$x=-7, 2, \frac{19}{3}$$

D

$$x=-3, \pm \frac{i}{2}$$

E

$$x=-4, 0, -1$$

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