How do you expand #(5y^4-x)^3#?

1 Answer
Oct 20, 2016

See below

Explanation:

Write it out:
#(5y^4 - x)(5y^4 - x)(5y^4 - x)# since it says to the power of 3

Step 1:
Pick either TWO brackets to expand it.
I would pick #(5y^4 - x)(5y^4 - x)#
This is how you expand them:

Cool Math
So, #(5y^4 xx 5y^4) + (5y^4 xx -x) + (-x xx 5y^4) + (-x xx -x)# should give you #25y^8 - 5xy^4 - 5xy^4 + x^2#.
BE VERY CAREFUL WITH NEGATIVES!!

Step 2: Now fully simplify the expanded equation
Simplify #25y^8 - 5xy^4 - 5xy^4 + x^2#.
Look out for like terms! There are two like #xy^4# terms. Add the like terms together. There is only one like term: which is #- 5xy^4 - 5xy^4#. So add these two up and it will become #-10xy^4#.
Your simplified equation should look like #25y^8 - 10xy^4 + x^2#.
BE VERY CAREFUL WITH NEGATIVES!!

Step 3: Now multiply your "expanded and simplified" two brackets with the remaining third bracket.
#(5y^4 - x)(25y^8 - 10xy^4 + x^2)#

Break the third bracket up (that you did not touch) into #5y^4# and #-x# and multiply each of it with the expanded brackets.
#[5y^4(25y^8 - 10xy^4 + x^2)] + [-x(25y^8 - 10xy^4 + x^2)]#

Step 4: Expand them and add them up.
BE VERY CAREFUL WITH NEGATIVES!!

Expand #5y^4(25y^8 - 10xy^4 + x^2)# and it will give you:
#125y^12 - 50xy^8 + 5x^2y^2#

Now expand #-x(25y^8 - 10xy^4 + x^2)# and it will give you:
#-25xy^8 + 10xy^4 - x^3#.

Now add them up:
#(125y^12 - 50xy^8 + 5x^2y^2) + (-25xy^8 + 10xy^4 - x^3)#
#= 125y^12 - 50xy^8 + 5x^2y^2 -25xy^8 + 10xy^4 - x^3#

Step 5: Last but not least, simplify them & add the like terms together.

# 125y^12 - 50xy^8 + 5x^2y^2 -25xy^8 + 10xy^4 - x^3#

Look closely at the equation and you can see one like term which is #xy^8#.

Add the like terms together.
#-50xy^8 - 25xy^8 = -75xy^8#

Your final equation should look like:
# 125y^12 - 75xy^8 + 5x^2y^2 + 10xy^4 - x^3#