Problem 106 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(b=1\).

Step 2: 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 \(x^{3} + 1^{3} = (x+1)(x^{2}-x*1+1^{2})\).

Step 3: Simplify the polynomial

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