Chapter 11: Q 265 (page 1069)

In the following exercises, factor completely.
$8 x^{3} - 27 y^{3}$

Short Answer

Expert verified

The expression is factored as $8 x^{3} - 27 y^{3} = (2 x - 3 y) (4 x^{2} + 6 x y + 9 y^{2})$

Step by step solution

Step 1. Given Information

Consider the expression $8 x^{3} - 27 y^{3}$

The objective is to factor the expression completely.

Step 2. Factor using the difference of cube.

The difference of cube can be factored as

$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$

In the given binomial $8 x^{3} - 27 y^{3}$ , the first term $8 x^{3}$ is the cube of $2 x$ and the second term $27 y^{3}$ is the cube of $3 y$ . So it can be factored using the difference of cube formula as

localid="1644651551111" $\begin{array}{rcl} 8 x^{3} - 27 y^{3} & = & {(2 x)}^{3} - {(3 y)}^{3} \\ = & (2 x - 3 y) {{(2 x)}^{2} + (2 x) (3 y) + {(3 y)}^{2}} \\ = & (2 x - 3 y) (4 x^{2} + 6 x y + 9 y^{2}) \end{array}$

Step 3. Check the factors

Multiply the factors to check the solution

$\begin{array}{rcl} (2 x - 3 y) (4 x^{2} + 6 x y + 9 y^{2}) & = & 2 x (4 x^{2} + 6 x y + 9 y^{2}) - 3 y (4 x^{2} + 6 x y + 9 y^{2}) \\ = & 8 x^{3} + 12 x^{2} y + 18 x y^{2} - 12 x^{2} y - 18 x y^{2} - 27 y^{3} \\ = & 8 x^{3} - 27 y^{3} \end{array}$

And we get the given expression. So the expression is correctly factored.

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In the following exercises, factor completely.
$8 x^{3} - 27 y^{3}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Factor using the difference of cube.

Step 3. Check the factors

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In the following exercises, factor completely. 8x3-27y3

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Factor using the difference of cube.

Step 3. Check the factors

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Discrete Mathematics

Applied Mathematics

Statistics

Calculus

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In the following exercises, factor completely.
$8 x^{3} - 27 y^{3}$