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Chapter 0: Problem 59

Factor using the formula for the sum or difference of two cubes. $$x^{3}-64$$

Short Answer

Expert verified
The factorization of \(x^{3}-64\) using the formula for the sum or difference of two cubes is \((x-4)(x^{2} + 4x + 16)\).

Step by step solution

01

Identify the Problem

Your task is to factorize the expression \(x^{3}-64\). It is important to recognize this as a difference of two cubes. To do so it's useful to remember that \(64\) is a perfect cube and can be written as \(4^{3}\). Therefore, the problem can be rewritten as \(x^{3}-4^{3}\).
02

Use the Formula for the Difference of Two Cubes

The formula for the difference of two cubes is \(a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})\). Apply this formula to \(x^{3}-4^{3}\) and replace \(a\) with \(x\) and \(b\) with \(4\). This results in \((x-4)((x)^{2} + (x)(4) + (4)^{2})\).
03

Simplify the Expression

We can simplify the expression \((x-4)((x)^{2} + (x)(4) + (4)^{2})\) to \((x-4)(x^{2} + 4x + 16)\).

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