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Chapter 5: Problem 51

Factor completely. $$ 54 a^{3}-16 b^{3} $$

Short Answer

Expert verified
The factored form is \( (3a - 2b)(9a^2 + 6ab + 4b^2) \).

Step by step solution

01

Recognize the form

The given expression is in the form of a difference of cubes: \[ 54a^3 - 16b^3 \] which can be factored using the formula for the difference of cubes: \[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]
02

Rewrite the expression

Rewrite the expression in terms of cubes: \[ (3a)^3 - (2b)^3 \]
03

Apply the difference of cubes formula

Apply the difference of cubes formula where \[ x = 3a \text{ and } y = 2b \]: \[ (3a)^3 - (2b)^3 = (3a - 2b)((3a)^2 + (3a)(2b) + (2b)^2) \]
04

Simplify the second part

Simplify the second part of the factorization: \[ (3a)^2 + (3a)(2b) + (2b)^2 = 9a^2 + 6ab + 4b^2 \]
05

Write the final factored form

Substitute back to get the full factored expression: \[ 54a^3 - 16b^3 = (3a - 2b)(9a^2 + 6ab + 4b^2) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Cubes //the value of concept will be the headline later on so DO NOT restate the headline in text
The 'difference of cubes' refers to any expression that can be written in the form \[ x^3 - y^3 \]. This form is important because it can be factored using a specific, straightforward formula.
To factor a difference of cubes, we use this formula: \[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \] The formula might look a bit complicated, but it's simpler when broken down.
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