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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. 6.64 (page 600)

Factor: $1000 m^{3} - 125 n^{3}$ .

Short Answer

Expert verified

Factors of $1000 m^{3} - 125 n^{3}$ are $(10 m - 5 n) (100 m^{2} + 50 m n + 25 n^{2})$ .

Step by step solution

01

Step 1. Given information.

Given an expression $1000 m^{3} - 125 n^{3}$ .

02

Step 2. Check if the binomial is a difference and write the terms as cubes.

Consider $a = 10 m$ and $b = 5 n$ .

It can be seen that $1000 m^{2} - 125 n^{2} = {(10 m)}^{2} - {(5 n)}^{2}$ .

Therefore, $1000 m^{3} - 125 n^{3}$ is a difference of 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 as follows.

$\begin{array}{rcl} 1000 m^{3} - 125 n^{2} & = & {(10 m)}^{3} - {(5 n)}^{2} \\ = & (10 m - 5 n) ({(10 m)}^{2} + 50 m n + {(5 n)}^{2}) \\ = & (10 m - 5 n) (100 m^{2} + 50 m n + 25 n^{2}) \end{array}$

04

Step 4. Check by multiplying the factors.

Verify as follows.

$\begin{array}{rcl} (10 m - 5 n) (100 m^{2} + 50 m n + 25 n^{2}) & = & 10 m (100 m^{2} + 50 m n + 25 n^{2}) - 5 n (100 m^{2} + 50 m n + 25 n^{2}) \\ = & 1000 m^{3} + 500 m^{2} n + 250 n^{2} m - 500 n m^{2} - 250 m n^{2} - 125 n^{3} \\ = & 1000 m^{3} - 125 n^{3} \end{array}$

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Most popular questions from this chapter

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