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All courses Math III · A-SSE.1.b 15 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. Enter the role and its applicable zero, quantity/unit, or positivity fields.
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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

Focus on the single factor

\[x-4\]

The question asks what the factor \((x-4)\) means by itself inside the factored expression.

02

Set the factor equal to zero

\[x-4~=~0\]

A factor creates a polynomial zero when that factor itself becomes zero.

03

Solve for x

\[x~=~4\]

So the condition attached to this factor is that \(x=4\) makes \((x-4)\) equal zero.

04

State the interpretation

\[x~=~4\Rightarrow~(x-4)~=~0\Rightarrow~P(x)~=~0\]

That means the factor \((x-4)\) represents the condition \(x=4\), which creates a zero of the polynomial.

05

State the final answer

\[\text{role}=\text{zero}-\text{producing}~\text{factor};~\text{zero}~\text{condition}=x-4=0;~\text{real}~\text{zero}=x=4;~\text{polynomial}~\text{result}=P(4)=0\]

The factor \((x-4)\) represents the condition \(x=4\), which makes that factor zero and creates a polynomial zero.

+

Another way

  1. You can substitute \(x=4\) into \(P(x)=(x-4)(x+2)\) and see that one factor becomes \(0\), so the whole product becomes \(0\).

!

Common mistake

A common mistake is to interpret \((x-4)\) as a length or context quantity here, but in this polynomial it is being used as a zero-producing factor.