Chapter 6: Q 6.68 (page 601)

Factor: ${(n + 3)}^{3} - 125 n^{3}$ .

Short Answer

Expert verified

Factors of ${(n + 3)}^{3} - 125 n^{3}$ are $(- 4 n + 3) (31 n^{2} + 21 n + 9)$ .

Step by step solution

Step 2. Given information.

Given an expression ${(n + 3)}^{3} - 125 n^{3}$ .

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

It can be seen that the binomial is a difference.

Consider $a = n + 3$ and $b = 5 n$ .

Then, ${(n + 1)}^{3} - 125 n^{3} = {(n + 1)}^{3} - {(5 n)}^{3}$ .

Therefore, the binomial is a difference of perfect cubes.

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.

$\begin{array}{rcl} {(n + 3)}^{3} - 125 n^{3} & = & {(n + 3)}^{3} - {(5 n)}^{3} \\ = & (n + 3 - 5 n) ({(n + 3)}^{2} + 5 n (n + 3) + {(5 n)}^{2}) \\ = & (- 4 n + 3) (n^{2} + 6 n + 9 + 5 n^{2} + 15 n + 25 n^{2}) \\ = & (- 4 n + 3) (31 n^{2} + 21 n + 9) \end{array}$

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.

$\begin{array}{rcl} (- 4 n + 3) (31 n^{2} + 21 n + 9) & = & - 4 n (31 n^{2} + 21 n + 9) + 3 (31 n^{2} + 21 n + 9) \\ = & - 124 n^{3} + 9 n^{2} + 27 n + 27 \\ = & n^{3} + 3^{3} + 3 {(n)}^{2} 3 + 3 n {(3)}^{2} - 125 n^{3} \\ = & {(n + 3)}^{3} - 125 n^{3} \end{array}$

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Factor: ${(n + 3)}^{3} - 125 n^{3}$ .

Short Answer

Step by step solution

Step 2. Given information.

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

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

Step 4. Check the result by multiplying the factors.

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Factor:n+33-125n3.

Short Answer

Step by step solution

Step 2. Given information.

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

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

Step 4. Check the result by multiplying the factors.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Probability and Statistics

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Study anywhere. Anytime. Across all devices.

Factor: ${(n + 3)}^{3} - 125 n^{3}$ .