Question

Quotient + remainder
I need help I would have 20 points

Answer 1

We want to compute

`(-5x^4+4x^3-20x^2+15x-16)/(-x^2-3)`

The goal is to write it in the following form:

`-5x^4+4x^3-20x^2+15x-16=Q(x)(-x^2-3)+R(x)`

First step: How many "copies" of
`-x^2` does
`-5x^4` contain? We can write
`-5x^4=(-x^2)(5x^2)`, so the answer is
`5x^2` copies.

If we distribute
`5x^2` to
`-x^2-3`, we get

`5x^2(-x^2-3)=-5x^4-15x^2`

If we were done, then this product would have matched the numerator at the start. But it doesn't; there are still some terms that need to vanish. In particular, we still have to account for

`(-5x^4+4x^3-20x^2+15x-16)-(-5x^4-15x^2)=4x^3-5x^2+15x-16`

Second step: How many copies of
`-x^2` can we find in
`4x^3`? Well,
`4x^3=(-4x)(-x^2)`, so the answer is
`-4x`. Then

`(5x^2-4x)(-x^2-3)=-5x^4+4x^3-15x^2+12x`

but this still doesn't match the original numerator. We still have to deal with

`(-5x^4+4x^3-20x^2+15x-16)-(5x^2-4x)(-x^2-3)=-5x^2+3x-16`

Third step: How many copies of
`-x^2` can we get out of
`-5x^2`?
`-5x^2=5(-x^2)`. Then

`(5x^2-4x+5)(-x^2-3)=-5x^4+4x^3-20x^2+12x-15`

This also doesn't match the original numerator. We have a difference between what we have and what we want of

`(-5x^4+4x^3-20x^2+15x-16)-(5x^2-4x+5)(-x^2-3)=3x-1`

But we also can't divide
`3x` into chunks of
`-x^2`, and we can't continue the division algorithm.

Our final answer would be

`(-5x^4+4x^3-20x^2+15x-16)/(-x^2-3)=5x^2-4x+5+(3x-1)/(-x^2-3)`

For the second problem, you should find

`(-15x^3-13x+4)/(5x^2+2)=-3x+(-7x+4)/(5x^2+2)`