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Chapter 0: Problem 106

Explain how to factor \(x^{3}+1\).

Short Answer

Expert verified
The factorization of \(x^{3}+1\) is \((x+1)(x^{2}-x+1)\).

Step by step solution

01

Identify the sum of cubes formula

The first step is to identify the sum of cubes formula. This is given as \(a^{3} + b^{3} = (a+b)(a^{2}-ab+b^{2})\). The polynomial \(x^{3}+1\) is a sum of cubes, where \(a=x\) and \(b=1\).
02

Apply the sum of cubes formula

This step involves applying the sum of cubes formula. Replace \(a\) with \(x\) and \(b\) with 1 in the formula. This gives \(x^{3} + 1^{3} = (x+1)(x^{2}-x*1+1^{2})\).
03

Simplify the polynomial

The last step is to simplify the polynomial. The result from step 2 simplifies to: \(x^{3} + 1 = (x+1)(x^{2}-x+1)\).

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