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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 6: Q 6.58 (page 597)

Factor: $7 a^{4} c^{2} - 7 b^{4} c^{2}$ .

Short Answer

Expert verified

Factors of $7 a^{4} c^{2} - 7 b^{4} c^{2}$ are $7 c^{2} (a - b) (a + b) (a^{2} + b^{2})$ .

Step by step solution

01

Step 1. Given information.

Given an expression $7 a^{4} c^{2} - 7 b^{4} c^{2}$ .

02

Step 2. Find the GCF of the expression.

Note that GCF stands for Greatest common factor.

In the expression $7 a^{4} c^{2} - 7 b^{4} c^{2}$ has $7 c^{2}$ as its GCF.

The expression can be written as $7 c^{2} (a^{4} - b^{4})$ .

03

Step 3. Find if the binomial is a difference of squares and factor as a product of conjugates.

Note that $x^{2} - y^{2} = (x - y) (x + y)$ .

The binomial can be expressed as a difference of squares as follows.

$\begin{array}{rcl} 7 c^{2} (a^{4} - b^{4}) & = & 7 c^{2} ({(a^{2})}^{2} - {(b^{2})}^{2}) \\ = & 7 c^{2} (a^{2} - b^{2}) (a^{2} + b^{2}) \end{array}$

04

Step 4. Check if the factors are also a difference of squares.

It can be seen that factor $a^{2} - b^{2}$ is a difference of squares.

Note that sum of squares cannot be factored.

Factor the result as the product of conjugates as follows.

$7 c^{2} (a^{2} - b^{2}) (a^{2} + b^{2}) = 7 c^{2} (a + b) (a - b) (a^{2} + b^{2})$

05

Step 5. Verify.

Multiply the factors as follows.

$\begin{array}{rcl} 7 c^{2} (a - b) (a + b) (a^{2} + b^{2}) & = & 7 c^{2} (a^{2} - b^{2}) (a^{2} + b^{2}) \\ = & 7 c^{2} (a^{4} - b^{4}) \\ = & 7 c^{2} a^{4} - 7 c^{2} b^{4} \end{array}$

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Most popular questions from this chapter

In the following exercises, factor the greatest common factor from each polynomial.

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Factor by Grouping In the following exercises, factor by grouping.

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