Simplify the following expression: \frac{(a+b)^3-3ab(a+b)}{(a+b)^2-2ab}

Question:

Simplify the following expression: {eq}\frac{(a+b)^3-3ab(a+b)}{(a+b)^2-2ab} {/eq}

Algebraic Expression:

A simplified form of an algebraic expression is that if the numerator and denominator don't have any common factor. Here, we use the two identities that are as follows :

{eq}\displaystyle (x +y)^2 = x^2 + y^2 + 2xy\\ \text{and}\\ \displaystyle (x +y)^3 = x^3 + y^3 + 3x^2y + 3xy^2\\ {/eq}

Answer and Explanation:

We are given that {eq}\displaystyle \frac{(a+b)^3-3ab(a+b)}{(a+b)^2-2ab} {/eq}

Using the Algebraic identities

{eq}\displaystyle (x +y)^2 = x^2 + y^2 + 2xy\\ \text{and}\\ \displaystyle (x +y)^3 = x^3 + y^3 + 3x^2y + 3xy^2\\ {/eq}

Thus,

{eq}\begin{align*} \displaystyle \frac{(a+b)^3-3ab(a+b)}{(a+b)^2-2ab} & = \displaystyle \frac{a^3 + b^3 + 3a^2b + 3ab^2-3ab(a+b)}{a^2 + b^2 + 2ab -2ab} \\ & = \displaystyle \frac{a^3 + b^3 + 3a^2b + 3ab^2-3a^2b-3ab^2}{a^2 + b^2} \\ & = \displaystyle \frac{a^3 + b^3}{a^2 + b^2} \\ \end{align*} {/eq}


Learn more about this topic:

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Evaluating Simple Algebraic Expressions

from ELM: CSU Math Study Guide

Chapter 6 / Lesson 3
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