Question

Use long division and divided (2x^5-15x^3-2x^2+10x-24) by (x^2-x-4)

Answer 1

`2x^5-15x^3-2x^2+10x-24=(x^2-x-4)(2x^3+2x^2-5x+1)+(-9x-20)`

Explanation:

Use long division and divided
`(2x^5-15x^3-2x^2+10x-24)` by
`(x^2-x-4).`

1. Multiply
`(x^2-x-4)` by
`2x^3` and subtract the result from
`(2x^5-15x^3-2x^2+10x-24):`

`2x^5-15x^3-2x^2+10x-24-2x^3(x^2-x-4)=\\ \\=2x^5-15x^3-2x^2+10x-24-2x^5+2x^4+8x^3=\\ \\=2x^4-7x^3-2x^2+10x-24`

2. Multiply
`(x^2-x-4)` by
`2x^2` and subtract the result from
`(2x^4-7x^3-2x^2+10x-24):`

`2x^4-7x^3-2x^2+10x-24-2x^2(x^2-x-4)=\\ \\=2x^4-7x^3-2x^2+10x-24-2x^4+2x^3+8x^2=\\ \\=-5x^3+6x^2+10x-24`

3. Multiply
`(x^2-x-4)` by
`-5x` and subtract the result from
`(-5x^3+6x^2+10x-24):`

`-5x^3+6x^2+10x-24-(-5x)(x^2-x-4)=\\ \\=-5x^3+6x^2+10x-24+5x^3-5x^2-20x=\\ \\=x^2-10x-24`

4. Multiply
`(x^2-x-4)` by
`1` and subtract the result from
`(x^2-10x-24):`

`x^2-10x-24-1\cdot (x^2-x-4)=\\ \\=x^2-10x-24-x^2+x+4=\\ \\=-9x-20`

5. The result of the long division is

`2x^5-15x^3-2x^2+10x-24=(x^2-x-4)(2x^3+2x^2-5x+1)+(-9x-20)`