Problem 114 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The factored form of \(x^{3}+1\) is \((x + 1)(x^{2} - x + 1)\)

Step 1: Identify a and b

In the given equation \(x^{3}+1\), \(a\) is \(x\) (since \(x^3\) is equivalent to \((x)^3\)) and \(b\) is 1 (since \(1\) is equivalent to \((1)^3\)).

Step 2: Apply the sum of cubes formula

Substitute \(a\) and \(b\) into the sum of cubes factoring formula \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\), which results in \((a + b)(a^2 - ab + b^2) = (x + 1)(x^2 - x*1 + 1^2)\)

Step 3: Simplify the equation

Simplify the equation to get a simplified factored expression. The equation simplifies to \((x + 1)(x^{2} - x + 1)\)