Q 6.58 - Intermediate Algebra

Question

Factor: $7a^4{c}^2-7b^4{c}^2$.

Step-by-Step Solution

Answer

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

Step 1. Given information.

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

Step 2. Find the GCF of the expression.

Note that GCF stands for Greatest common factor.

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

The expression can be written as $7c^2\left(a^4-b^4\right)$.

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

Note that $x^2-y^2=\left(x-y\right)\left(x+y\right)$.

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

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

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.

$7c^2\left(a^2-b^2\right)\left(a^2+b^2\right)=7c^2\left(a+b\right)\left(a-b\right)\left(a^2+b^2\right)$

Step 5. Verify.

Multiply the factors as follows.

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