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.
• 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
From here, we can use synthetic division, along with our upper and lower bound rules to narrow down the possibilities. Eventually, we will find that 5 is a zero: $$\frac{x^3-3x^2-12x+10}{x-5}=x^2 + 2x - 2$$ We can use this to factor our polynomial: $$f(x)=(x - 5)(x^2 + 2x - 2)$$ We can take the quadratic factor and use our quadratic formula to obtain: $$x=-1 \pm \sqrt{3}$$ This gives us the following zeros for our polynomial function: $$x=-1 \pm \sqrt{3}, 5$$

Skills Check:

Example #1

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

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

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