Question

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

Answer 1

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:

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`