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] using long division, we proceed as follows:
1. Set up the division as a long division problem similar to numerical long division.
2. Divide the leading term of the dividend by the leading term of the divisor.
- Divide [tex]\(-3x^5\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(-3x^2\)[/tex].
3. Multiply the entire divisor [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex] by [tex]\(-3x^2\)[/tex]:
- [tex]\(-3x^2 \cdot (x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2\)[/tex].
4. Subtract the result from the original dividend:
- [tex]\((-3x^5 + 11x^4 + 33x^3 - 26x^2 - 36x - 6) - (-3x^5 - 18x^4 + 9x^3 + 15x^2)\)[/tex]
- [tex]\(= 11x^4 + 18x^4 + 33x^3 - 9x^3 - 26x^2 - 15x^2 - 36x - 6\)[/tex]
- [tex]\(= 29x^4 + 24x^3 - 41x^2 - 36x - 6\)[/tex].
5. Repeat the process dividing the new leading term by the leading term of the divisor:
- Divide [tex]\(29x^4\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(29x\)[/tex].
6. Multiply the entire divisor [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex] by [tex]\(29x\)[/tex]:
- [tex]\(29x \cdot (x^3 + 6x^2 - 3x - 5) = 29x^4 + 174x^3 - 87x^2 - 145x\)[/tex].
7. Subtract the result from the previous remainder:
- [tex]\((29x^4 + 24x^3 - 41x^2 - 36x - 6) - (29x^4 + 174x^3 - 87x^2 - 145x)\)[/tex]
- [tex]\(= 24x^3 - 174x^3 - 41x^2 + 87x^2 - 36x + 145x - 6\)[/tex]
- [tex]\(= -150x^3 + 46x^2 + 109x - 6\)[/tex].
8. Repeat the process once more with the new leading term:
- Divide [tex]\(-150x^3\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(-150\)[/tex].
9. Multiply the entire divisor [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex] by [tex]\(-150\)[/tex]:
- [tex]\(-150 \cdot (x^3 + 6x^2 - 3x - 5) = -150x^3 - 900x^2 + 450x + 750\)[/tex].
10. Subtract the final result from the last remainder:
- [tex]\((-150x^3 + 46x^2 + 109x - 6) - (-150x^3 - 900x^2 + 450x + 750)\)[/tex]
- [tex]\(= 46x^2 + 900x^2 + 109x - 450x - 6 - 750\)[/tex]
- [tex]\(= 946x^2 - 341x - 756\)[/tex].
The quotient is [tex]\(-3x^2 + 29x - 150\)[/tex] and the remainder is [tex]\(946x^2 - 341x - 756\)[/tex].
So the result of the division is:
[tex]\[
\frac{-3x^5 + 11x^4 + 33x^3 - 26x^2 - 36x - 6}{x^3 + 6x^2 - 3x - 5} = -3x^2 + 29x - 150 + \frac{946x^2 - 341x - 756}{x^3 + 6x^2 - 3x - 5}
\][/tex]
