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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. 6.67 (page 601)

Factor: ${(y + 1)}^{3} - 27 y^{3}$ .

Short Answer

Expert verified

Factors of ${(y + 1)}^{3} - 27 y^{3}$ are $(- 2 y + 1) (13 y^{2} + 5 y + 1)$ .

Step by step solution

01

Step 1. Given information.

Given an expression ${(y + 1)}^{3} - 27 y^{3}$ .

02

Step 2. Check if the binomial is a difference of perfect cubes.

It can be seen that the binomial is a difference.

Consider $a = y + 1$ and $b = 3 y$ .

Then, ${(y + 1)}^{3} - 27 y^{3} = {(y + 1)}^{3} - {(3 y)}^{3}$ .

Therefore, the binomial is a difference of perfect cubes.

03

Step 3. Use the difference of cubes pattern and simplify.

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

Factor the expression and simplify as follows.

$\begin{array}{rcl} {(y + 1)}^{3} - 27 y^{3} & = & {(y + 1)}^{3} - {(3 y)}^{3} \\ = & (y + 1 - 3 y) ({(y + 1)}^{2} + 3 y (y + 1) + {(3 y)}^{2}) \\ = & (- 2 y + 1) (y^{2} + 2 y + 1 + 3 y^{2} + 3 y + 9 y^{2}) \\ = & (- 2 y + 1) (13 y^{2} + 5 y + 1) \end{array}$

04

Step 4. Check the result by multiplying the factors.

Note that ${(a + b)}^{3} = a^{3} + b^{3} + 3 a^{2} b + 3 a b^{2}$ .

Verify the result as follows.

$\begin{array}{rcl} (- 2 y + 1) (13 y^{2} + 5 y + 1) & = & - 2 y (13 y^{2} + 5 y + 1) + 1 (13 y^{2} + 5 y + 1) \\ = & - 26 y^{3} + 3 y^{2} + 3 y + 1 \\ = & y^{3} + 1^{3} + 3 y^{2} (1) + 3 y {(1)}^{2} - 27 y^{3} \\ = & {(y + 1)}^{3} - 27 y^{3} \end{array}$

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Most popular questions from this chapter

Factor completely using trial and error: $2 a^{2} + 5 a + 3$

In the following exercises, factor each trinomial of the form $x^{2} + b x + c$ .

$b^{2} + 14 b + 48$

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

$14 p + 35$

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error.

$5 y^{2} + 16 y + 11$

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.
$z^{6} - 27$

See all solutions

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