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

Oct 20, 2016

See below

#### Explanation:

Write it out:
$\left(5 {y}^{4} - x\right) \left(5 {y}^{4} - x\right) \left(5 {y}^{4} - x\right)$ since it says to the power of 3

Step 1:
Pick either TWO brackets to expand it.
I would pick $\left(5 {y}^{4} - x\right) \left(5 {y}^{4} - x\right)$
This is how you expand them:

So, $\left(5 {y}^{4} \times 5 {y}^{4}\right) + \left(5 {y}^{4} \times - x\right) + \left(- x \times 5 {y}^{4}\right) + \left(- x \times - x\right)$ should give you $25 {y}^{8} - 5 x {y}^{4} - 5 x {y}^{4} + {x}^{2}$.
BE VERY CAREFUL WITH NEGATIVES!!

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

Step 3: Now multiply your "expanded and simplified" two brackets with the remaining third bracket.
$\left(5 {y}^{4} - x\right) \left(25 {y}^{8} - 10 x {y}^{4} + {x}^{2}\right)$

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

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

Expand $5 {y}^{4} \left(25 {y}^{8} - 10 x {y}^{4} + {x}^{2}\right)$ and it will give you:
$125 {y}^{12} - 50 x {y}^{8} + 5 {x}^{2} {y}^{2}$

Now expand $- x \left(25 {y}^{8} - 10 x {y}^{4} + {x}^{2}\right)$ and it will give you:
$- 25 x {y}^{8} + 10 x {y}^{4} - {x}^{3}$.

$\left(125 {y}^{12} - 50 x {y}^{8} + 5 {x}^{2} {y}^{2}\right) + \left(- 25 x {y}^{8} + 10 x {y}^{4} - {x}^{3}\right)$
$= 125 {y}^{12} - 50 x {y}^{8} + 5 {x}^{2} {y}^{2} - 25 x {y}^{8} + 10 x {y}^{4} - {x}^{3}$

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

$125 {y}^{12} - 50 x {y}^{8} + 5 {x}^{2} {y}^{2} - 25 x {y}^{8} + 10 x {y}^{4} - {x}^{3}$

Look closely at the equation and you can see one like term which is $x {y}^{8}$.

$- 50 x {y}^{8} - 25 x {y}^{8} = - 75 x {y}^{8}$
$125 {y}^{12} - 75 x {y}^{8} + 5 {x}^{2} {y}^{2} + 10 x {y}^{4} - {x}^{3}$