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

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Answer 1 · 2018-01-16T21:10:58

See a solution process below:

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

#a^3 - a^2b + ab^2 - a^2b + ab^2 - b^3#

Next, group like terms:

#a^3 - a^2b - a^2b + ab^2 + ab^2 - b^3#

Now, combine like terms:

#a^3 - 1a^2b - 1a^2b + 1ab^2 + 1ab^2 - b^3#

#a^3 + (-1 - 1)a^2b + (1 + 1)ab^2 - b^3#

#a^3 + (-2)a^2b + 2ab^2 - b^3#

#a^3 - 2a^2b + 2ab^2 - b^3#

Answer 2 · 2018-01-16T21:19:43

#=a^3-2a^2b+2ab^2-b^3#

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

Given: #(a^3-a^2b+ab^2)+(-a^2b+ab^2-b^3)#

We can highlight the common terms...

#(a^3color(red)(-a^2b)+color(blue)(ab^2))+(color(red)(-a^2b)+color(blue)(ab^2)-b^3)#

...And combine them

#color(red)(-a^2b)+color(red)(-a^2b)=color(red)(-a^2b-a^2b)=color(red)(-2a^2b)#

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

So simplifying the problem we get:

#=a^3color(red)(-2a^2b)+color(blue)(2ab^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