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Chapter 5: Problem 133

Factor each polynomial by grouping. $$ x^{3}+7 x-7 a-a x^{2} $$

Short Answer

Expert verified
Factor: \((x - a)(x^2 + 7)\).

Step by step solution

01

Rewrite the polynomial

Group the polynomial into two pairs of terms to prepare for factoring by grouping. Rewrite the given polynomial: \(x^3 + 7x - 7a - ax^2\). Arrange terms in a way that will facilitate grouping: \(x^3 - ax^2 + 7x - 7a\).
02

Factor out the greatest common factor

From each group, factor out the greatest common factor (GCF). For the first group \(x^3 - ax^2\), factor out \(x^2\): \(x^2 (x - a)\). For the second group \(7x - 7a\), factor out \(7\): \(7 (x - a)\).
03

Combine the factored expressions

Now put the factored terms together: \(x^2 (x - a) + 7 (x - a)\).
04

Factor out the common binomial factor

Notice that \(x - a\) is a common factor. Factor \(x - a\) out from each term: \((x - a)(x^2 + 7)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring by Grouping
Factoring by grouping is a method used when you have a polynomial that can be divided into two or more groups. Each group will have a common factor, making it easier to factor the entire polynomial. It's especially useful when the polynomial has four terms.

The first step is to rearrange the terms if necessary, to make grouping easier. For example, given the polynomial: ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline
Greatest Common Factor
Once grouped, the next step in factoring by grouping is to factor out the greatest common factor (GCF) from each group. The GCF is the largest factor that two or more terms share.

For instance, in the group ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline
Binomial Factor
Finally, after factoring out the GCF from each group, you look for a common binomial factor. A binomial factor is a two-term expression that is common to both groups.

For example, in the polynomial, ewline (x - a)ewline The binomial factor is (x - a) because it appears in both groups. Factoring this out: ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline

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