# How do you combine (a ^ { 3} - a ^ { 2} b + a b ^ { 2} ) + ( - a ^ { 2} b + a b ^ { 2} - b ^ { 3} )?

Jan 16, 2018

See a solution process below:

#### Explanation:

First, remove all of the terms from parenthesis. Be careful to handle the signs of each individual term correctly:

${a}^{3} - {a}^{2} b + a {b}^{2} - {a}^{2} b + a {b}^{2} - {b}^{3}$

Next, group like terms:

${a}^{3} - {a}^{2} b - {a}^{2} b + a {b}^{2} + a {b}^{2} - {b}^{3}$

Now, combine like terms:

${a}^{3} - 1 {a}^{2} b - 1 {a}^{2} b + 1 a {b}^{2} + 1 a {b}^{2} - {b}^{3}$

${a}^{3} + \left(- 1 - 1\right) {a}^{2} b + \left(1 + 1\right) a {b}^{2} - {b}^{3}$

${a}^{3} + \left(- 2\right) {a}^{2} b + 2 a {b}^{2} - {b}^{3}$

${a}^{3} - 2 {a}^{2} b + 2 a {b}^{2} - {b}^{3}$

Jan 16, 2018

$= {a}^{3} - 2 {a}^{2} b + 2 a {b}^{2} - {b}^{3}$

#### Explanation:

Identify like/common terms first: (Terms with the same color are common terms that we can combine)

Given: $\left({a}^{3} - {a}^{2} b + a {b}^{2}\right) + \left(- {a}^{2} b + a {b}^{2} - {b}^{3}\right)$

We can highlight the common terms...

$\left({a}^{3} \textcolor{red}{- {a}^{2} b} + \textcolor{b l u e}{a {b}^{2}}\right) + \left(\textcolor{red}{- {a}^{2} b} + \textcolor{b l u e}{a {b}^{2}} - {b}^{3}\right)$

...And combine them

$\textcolor{red}{- {a}^{2} b} + \textcolor{red}{- {a}^{2} b} = \textcolor{red}{- {a}^{2} b - {a}^{2} b} = \textcolor{red}{- 2 {a}^{2} b}$

color(blue)(ab^2)+color(blue)(ab^2)=color(blue)(2ab^2

So simplifying the problem we get:

$= {a}^{3} \textcolor{red}{- 2 {a}^{2} b} + \textcolor{b l u e}{2 a {b}^{2}} - {b}^{3}$

Note: We did not combine ${a}^{3}$ and $- {b}^{3}$ because there are no other common terms to combine them with so we left them as is