Chapter 6: Q 6.57 (page 597)

Factor: $2 x^{4} y^{2} - 32 y^{2}$ .

Short Answer

Expert verified

Factors of $2 x^{4} y^{2} - 32 y^{2}$ are $2 y^{2} (x - 2) (x + 2) (x^{2} + 4)$ .

Step by step solution

Step 1. Given information.

Given an expression $2 x^{4} y^{2} - 32 y^{2}$ .

Step 2. Find the GCF of the expression.

Note that GCF stands for Greatest common factor.

In the expression $2 x^{4} y^{2} - 32 y^{2}$ has $2 y^{2}$ as its GCF.

The expression can be written as $2 y^{2} (x^{4} - 16)$ .

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

Note that $a^{2} - b^{2} = (a - b) (a + b)$ .

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

role="math" localid="1645275812595" $\begin{array}{rcl} 2 y^{2} (x^{4} - 16) & = & 2 y^{2} ({(x^{2})}^{2} - 4^{2}) \\ = & 2 y^{2} (x^{2} - 4) (x^{2} + 4) \end{array}$

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

It can be seen that factor $x^{2} - 4$ is a difference of squares.

Note that sum of squares cannot be factored.

Factor the result as the product of conjugates as follows.

$\begin{array}{rcl} 2 y^{2} (x^{2} - 4) (x^{2} + 4) & = & 2 y^{2} (x^{2} - 2^{2}) (x^{2} + 4) \\ = & 2 y^{2} (x - 2) (x + 2) (x^{2} + 4) \end{array}$

Step 5. Verify.

Multiply the factors as follows.

$\begin{array}{rcl} 2 y^{2} (x - 2) (x + 2) (x^{2} + 4) & = & 2 y^{2} (x^{2} - 2^{2}) (x^{2} + 4) \\ = & 2 y^{2} (x^{4} - 2^{4}) \\ = & 2 x^{4} y^{2} - 32 y^{2} \end{array}$

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Factor: $2 x^{4} y^{2} - 32 y^{2}$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the GCF of the expression.

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

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

Step 5. Verify.

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Factor:2x4y2-32y2.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the GCF of the expression.

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

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

Step 5. Verify.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Decision Maths

Geometry

Logic and Functions

Probability and Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Factor: $2 x^{4} y^{2} - 32 y^{2}$ .