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All courses Math III · A-APR.3 3 of 55
Identify zeros from factorizations and use them to sketch polynomial graphs.

Find zeros from a factored polynomial by setting each factor equal to zero

Problem
What are the zeros of this polynomial from its linear factors? (x - 3)(x + 2)(2x - 1)
Your answer

Hint

Set each linear factor equal to \(0\) one at a time.

A zero happens where a factor is \(0\). So solve \(x-3=0\), \(x+2=0\), and \(2x-1=0\).

Solution walkthrough

01

Use the factor x - 3

\[x~-~3~=~0,~\text{so}~x~=~3\]

If this factor equals \(0\), then the whole product equals \(0\), so \(3\) is a zero.

02

Use the factor x + 2

\[x~+~2~=~0,~\text{so}~x~=~-2\]

Subtracting \(2\) from both sides gives the input that makes this factor equal \(0\).

03

Use the factor 2x - 1

\[2x~-~1~=~0,~\text{so}~2x~=~1~\text{and}~x~=~1/2\]

Solving the last factor gives the zero \(1/2\).

04

State all zeros

\[\text{zeros}:~3,~-2,~1/2\]

Each linear factor gives one zero, so the polynomial has zeros \(3\), \(-2\), and \(1/2\).

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Another way

  1. You can use the Zero Product Property in reverse: a product is \(0\) exactly when at least one factor is \(0\).

!

Common mistake

A common mistake is to keep the same sign from the factor and say \(x + 2\) gives \(2\). The zero is the value that makes the factor equal \(0\), so \(x + 2 = 0\) gives \(-2\).