Chapter 8.7, Problem 54PFA

a.

To determine

To check if the polynomial is a difference of two squares.

Answer to Problem 54PFA

Yes

Explanation of Solution

Given:

Expression: $2x^2{y}^4-32x^2$

Concept used:

Difference of squares: $a^2-b^2=(a+b)(a-b)$

Calculation:

Consider the given equation,

  $\Rightarrow 2x^2{y}^4-32x^2$

Trying to apply difference of squares,

  $\begin{array}{l}\Rightarrow 2x^2{y}^4-32x^2\\ $ \Rightarrow 2({(x{y}^2)}^2-{(4x)}^2)$\Rightarrow 2(x{y}^2-4x)(x{y}^2+4x)\end{array}$

Conclusion:

Therefore, the given expression is a difference of squares.

b.

To determine

To factor the given expression.

Answer to Problem 54PFA

  $2(x{y}^2-4x)(x{y}^2+4x)$

Explanation of Solution

Given:

Expression: $2x^2{y}^4-32x^2$

Concept used:

Difference of squares: $a^2-b^2=(a+b)(a-b)$

Calculation:

Consider the given equation,

  $\Rightarrow 2x^2{y}^4-32x^2$

Trying to apply difference of squares,

  $\begin{array}{l}\Rightarrow 2x^2{y}^4-32x^2\\ $ \Rightarrow 2({(x{y}^2)}^2-{(4x)}^2)$\Rightarrow 2(x{y}^2-4x)(x{y}^2+4x)\end{array}$

Conclusion:

Therefore, the factor of the expression is $2(x{y}^2-4x)(x{y}^2+4x).$

c.

To determine

To explain the pattern used to factor.

Answer to Problem 54PFA

Difference of squares

Explanation of Solution

Given:

From subpart (a),

  $2(x{y}^2-4x)(x{y}^2+4x)$

Calculation:

The pattern used is difference of squares.

Conclusion:

Difference of squares is used.

d.

To determine

To write in a completely factored form.

Answer to Problem 54PFA

  $2x(y+2)(y-2)(y^2+4)$

Explanation of Solution

Given:

Expression: $2x^2{y}^4-32x^2$

Concept used:

Difference of squares

Calculation:

Therefore, from subpart (b),

  $\begin{array}{l}\Rightarrow 2(x{y}^2-4x)(x{y}^2+4x)\\ \Rightarrow 2x(y^2-4)(y^2+4)\end{array}$

Applying difference of squares to $(y^2-4)$

  $(y^2-4)=({(y)}^2-2^2)=(y+2)(y-2)$

The completely factored form is $2x(y+2)(y-2)(y^2+4)$

Conclusion:

The completely factored form is $2x(y+2)(y-2)(y^2+4).$