Question

-3x⁵+11x⁴+33x³-26x²-36x-6 / x³+6x²-3x-5​

Answer 1

`-3x^2+29x-150+(946x^2-341x-756)/(x^3+6x^2-3x-5)`

Explanation:

You have to divide the polynomial
`-3x^5+11x^4+33x^3-26x^2-36x-6` by the polynomial
`x^3+6x^2-3x-5:`

First, multiply the polynomial
`x^3+6x^2-3x-5` by
`-3x^2` and subtract the result from the polynomial
`-3x^5+11x^4+33x^3-26x^2-36x-6:`

`-3x^5+11x^4+33x^3-26x^2-36x-6-(-3x^2)(x^3+6x^2-3x-5)\\ \\=-3x^5+11x^4+33x^3-26x^2-36x-6+3x^5+18x^4-9x^3-15x^2\\ \\=29x^4+24x^3-41x^2-36x-6`

Now, multiply the polynomial
`x^3+6x^2-3x-5` by
`29x` and subtract the result from the polynomial
`29x^4+24x^3-41x^2-36x-6:`

`29x^4+24x^3-41x^2-36x-6-29x(x^3+6x^2-3x-5)\\ \\=29x^4+24x^3-41x^2-36x-6-29x^4-174x^3+87x^2+145x\\ \\=-150x^3+46x^2+109x-6`

At last, multiply the polynomial
`x^3+6x^2-3x-5` by
`-150` and subtract the result from the polynomial
`-150x^3+46x^2-181x-6:`

`-150x^3+46x^2+109x-6-(-150)(x^3+6x^2-3x-5)\\ \\=-150x^3+46x^2+109x-6+150x^3+900x^2-450x-750\\ \\=946x^2-341x-756`

So,

`(-3x^5+11x^4+33x^3-26x^2-36x-6)/(x^3+6x^2-3x-5)=\\ \\=((-3x^2+29x-150)(x^3+6x^2-3x-5)+946x^2-341x-756)/(x^3+6x^2-3x-5)=\\ \\=-3x^2+29x-150+(946x^2-341x-756)/(x^3+6x^2-3x-5)`