Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Our platform offers a seamless experience for finding reliable answers from a network of experienced professionals. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
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]
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]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
