Question

What is `(80x^4 + 48x^3 + 80x^2) -: 8x`?

Answer 1

`(80x^4 + 48x^3 + 80x^2)/(8x) = 10x^3 + 6x^2 + 10x`

Explanation:

Starting with: `(80x^4 + 48x^3 + 80x^2)/(8x)`

Dividing something by 8x is the same as multiplying it by `(8x)^-1`. Thus we have:
`(80x^4 + 48x^3 + 80x^2)(8x)^-1`

Notice that each of the terms is a factor of 8:
`((8^1)(10)x^4 + (8^1)(6)x^3 + (8^1)(10)x^2)(8^-1x^-1)`
Now multiply out the outer brackets and use the property `(x^a)(x^b)=x^(a+b)`:
`(8^(1-1))(10)x^(4-1) + (8^(1-1))(6)x^(3-1) + (8^(1-1)(10)x^(2-1))`
`(8^0)(10)x^3 + (8^0)(6)x^2 + (8^0)(10)x^1)`

Noting that `x^0 = 1 and x^1=x`:
`(10)x^3 + (6)x^2 + (10)x^1`
`10x^3 + 6x^2 + 10x`

If you didn't spot that all terms were a factor of 8, you could also say that `8^-1` = 1/8 = 0.125 and multiply each of the terms in the left bracket by `0.125/x`.

Alternatively, you may find it easier to split the terms up:
`((80x^4)/(8x) + (48x^3)/(8x) + (80x^2)/(8x)) = 10x^3 + 6x^2 + 10x`