Answer:
[tex]-3x^2+29x-150+\dfrac{946x^2-341x-756}{x^3+6x^2-3x-5}[/tex]
Step-by-step explanation:
You have to divide the polynomial [tex]-3x^5+11x^4+33x^3-26x^2-36x-6[/tex] by the polynomial [tex]x^3+6x^2-3x-5:[/tex]
First, multiply the polynomial [tex]x^3+6x^2-3x-5[/tex] by [tex]-3x^2[/tex] and subtract the result from the polynomial [tex]-3x^5+11x^4+33x^3-26x^2-36x-6:[/tex]
[tex]-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[/tex]
Now, multiply the polynomial [tex]x^3+6x^2-3x-5[/tex] by [tex]29x[/tex] and subtract the result from the polynomial [tex]29x^4+24x^3-41x^2-36x-6:[/tex]
[tex]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[/tex]
At last, multiply the polynomial [tex]x^3+6x^2-3x-5[/tex] by [tex]-150[/tex] and subtract the result from the polynomial [tex]-150x^3+46x^2-181x-6:[/tex]
[tex]-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[/tex]
So,
[tex]\dfrac{-3x^5+11x^4+33x^3-26x^2-36x-6}{x^3+6x^2-3x-5}=\\ \\=\dfrac{(-3x^2+29x-150)(x^3+6x^2-3x-5)+946x^2-341x-756}{x^3+6x^2-3x-5}=\\ \\=-3x^2+29x-150+\dfrac{946x^2-341x-756}{x^3+6x^2-3x-5}[/tex]
