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All courses Algebra II · A-APR.3 10 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
Find the zeros of \((x~-~3)(x~+~2)(2x~-~1)\).
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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

Apply the zero-product property

\[ABC=0~->~A=0~\text{or}~B=0~\text{or}~C=0\]

A product equals zero when at least one factor equals zero. Therefore set each of the three linear factors equal to zero and solve each equation separately.

02

Solve the monic factors

\[x-3=0~->~x=3;~x+2=0~->~x=-2\]

Adding 3 solves the first equation, and subtracting 2 solves the second. These yield zeros 3 and -2, with the signs determined by the equations rather than copied from the factors.

03

Solve the coefficiented factor

\[2x-1=0~->~2x=1~->~x=1/2\]

Add 1 to both sides and divide by the coefficient 2. The third zero is one-half, not 2.

04

Check and state all zeros

\[3,~-2,~1/2\]

At x=3 the first factor is zero; at x=-2 the second is zero; at x=1/2 the third is zero. Each input therefore makes the whole product zero. The zeros are 3, -2, and 1/2, choice A.

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Another way

  1. Rewrite each factor in the form coefficient times (x-zero): x-3, x-(-2), and 2(x-1/2), then read the zeros.

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Common mistake

Do not read factor 2x-1 as zero 2. Set it equal to zero and solve: 2x=1, so x=1/2.

Solution walkthrough video