Problem 122 - Introductory Algebra for College Students

Question

Factor completely. $$(y+1)^{3}+1$$

Step-by-Step Solution

Answer

The completely factorized form of \((y+1)^{3}+1\) is \((y+2)(y^2+y+2)\).

Step 1: Express the expression in the sum of cubes form

Identify \(a\) and \(b\) in your equation \(a^3 + b^3 = (y+1)^3 + 1\), so here \(a = (y+1)\) and \(b = 1\).

Step 2: Apply the sum of cubes formula

Substitute the values of \(a\) and \(b\) into the sum of cubes formula \(a^3 + b^3 = (a+b)(a^2-ab+b^2)\), so it becomes \((y+1)^3 + 1^3 = ((y+1)+1)[(y+1)^2-(y+1)*1+1^2]\).

Step 3: Simplify

Simplify the equation to get the factorized form. This becomes \((y+2)((y+1)^2-(y+1)+1) = (y+2)(y^2+2y+1-y+1) = (y+2)(y^2+y+2)\).