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All courses Algebra II · A-SSE.1.b 22 of 55
Interpret complex polynomial/rational expressions by treating parts as single units.

Classify the role of a factor as one unit

Problem
In \(P(x)~=~(x~-~4)(x~+~2)\), classify the factor \(x~-~4\). State its role, the relevant test and result, and any justified contextual constraint.
Your answer
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Hint

Identify the expression type, then test the factor for zero or connect it to the model's units.

A factor may create a zero, represent a contextual dimension, or remain positive while changing product values.

Solution walkthrough

01

Use the factor's position

\[P(x)=(x-4)(x+2):~x-4~\text{is}~a~\text{polynomial}~\text{factor}\]

The expression is a product defining a polynomial, so this factor can force the product to zero.

02

Test the factor for zero

\[x-4=0~->~x=4\]

Solving the complete factor equal to zero gives its associated real zero.

03

Verify the product output

\[P(4)=(4-4)(4+2)=0*6=0\]

At x equals 4, the tested factor is zero, so the entire product is zero.

04

Classify only what is justified

\[\text{Role}:~\text{zero}-\text{producing}~\text{polynomial}~\text{factor}.~\text{Test}:~x-4=0.~\text{Result}:~\text{real}~\text{zero}~x=4~\text{and}~P(4)=0.~\text{Context}:~\text{no}~\text{dimensional}~\text{constraint}~\text{is}~\text{given}.\]

The prompt supplies no length, time, or other modeled quantity, so no extra contextual domain restriction is warranted, choice A.

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Another way

  1. Use the zero-product property on both factors, then isolate the requested factor's contribution x equals 4.

!

Common mistake

Do not invent a positivity or dimension constraint when the expression is given only as a polynomial.

Solution walkthrough video