Answer :
To solve the problem of dividing the polynomial [tex]\(20x^5 - 24x^4 - 239x^3 + 201x^2 + 531x + 135\)[/tex] by [tex]\(2x^2 + x - 15\)[/tex], we'll perform polynomial division. Here’s a step-by-step guide:
1. Set Up the Division:
- We write the dividend [tex]\(20x^5 - 24x^4 - 239x^3 + 201x^2 + 531x + 135\)[/tex] and the divisor [tex]\(2x^2 + x - 15\)[/tex].
2. Divide the Leading Terms:
- Divide the leading term of the dividend ([tex]\(20x^5\)[/tex]) by the leading term of the divisor ([tex]\(2x^2\)[/tex]).
- [tex]\(20x^5 \div 2x^2 = 10x^3\)[/tex]. This will be the first term of our quotient.
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\(2x^2 + x - 15\)[/tex] by [tex]\(10x^3\)[/tex] to get:
[tex]\((10x^3)(2x^2 + x - 15) = 20x^5 + 10x^4 - 150x^3\)[/tex].
- Subtract this result from the original polynomial:
[tex]\((20x^5 - 24x^4 - 239x^3) - (20x^5 + 10x^4 - 150x^3) = -34x^4 - 89x^3\)[/tex].
4. Repeat This Process:
- Divide [tex]\(-34x^4\)[/tex] (new leading term) by [tex]\(2x^2\)[/tex]:
[tex]\(-34x^4 \div 2x^2 = -17x^2\)[/tex]. Add [tex]\(-17x^2\)[/tex] to the quotient.
- Multiply and subtract again:
- Multiply: [tex]\((-17x^2)(2x^2 + x - 15) = -34x^4 - 17x^3 + 255x^2\)[/tex].
- Subtract: [tex]\((-34x^4 - 89x^3 + 201x^2) - (-34x^4 - 17x^3 + 255x^2) = -72x^3 - 54x^2\)[/tex].
5. Continue the Process:
- Divide [tex]\(-72x^3\)[/tex] by [tex]\(2x^2\)[/tex]:
[tex]\(-72x^3 \div 2x^2 = -36x\)[/tex]. Add [tex]\(-36x\)[/tex] to the quotient.
- Multiply and subtract:
- Multiply: [tex]\((-36x)(2x^2 + x - 15) = -72x^3 - 36x^2 + 540x\)[/tex].
- Subtract: [tex]\((-72x^3 - 54x^2 + 531x) - (-72x^3 - 36x^2 + 540x) = -18x^2 - 9x\)[/tex].
6. Final Steps:
- Divide [tex]\(-18x^2\)[/tex] by [tex]\(2x^2\)[/tex]:
[tex]\(-18x^2 \div 2x^2 = -9\)[/tex]. Add [tex]\(-9\)[/tex] to the quotient.
- Multiply and subtract:
- Multiply: [tex]\((-9)(2x^2 + x - 15) = -18x^2 - 9x + 135\)[/tex].
- Subtract: [tex]\((-18x^2 - 9x + 135) - (-18x^2 - 9x + 135) = 0\)[/tex].
Thus, the quotient of the division is [tex]\(10x^3 - 17x^2 - 36x - 9\)[/tex] with a remainder of [tex]\(0\)[/tex].
So, the solution to the polynomial division is:
[tex]\[
\frac{20x^5 - 24x^4 - 239x^3 + 201x^2 + 531x + 135}{2x^2 + x - 15} = 10x^3 - 17x^2 - 36x - 9
\][/tex]
1. Set Up the Division:
- We write the dividend [tex]\(20x^5 - 24x^4 - 239x^3 + 201x^2 + 531x + 135\)[/tex] and the divisor [tex]\(2x^2 + x - 15\)[/tex].
2. Divide the Leading Terms:
- Divide the leading term of the dividend ([tex]\(20x^5\)[/tex]) by the leading term of the divisor ([tex]\(2x^2\)[/tex]).
- [tex]\(20x^5 \div 2x^2 = 10x^3\)[/tex]. This will be the first term of our quotient.
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\(2x^2 + x - 15\)[/tex] by [tex]\(10x^3\)[/tex] to get:
[tex]\((10x^3)(2x^2 + x - 15) = 20x^5 + 10x^4 - 150x^3\)[/tex].
- Subtract this result from the original polynomial:
[tex]\((20x^5 - 24x^4 - 239x^3) - (20x^5 + 10x^4 - 150x^3) = -34x^4 - 89x^3\)[/tex].
4. Repeat This Process:
- Divide [tex]\(-34x^4\)[/tex] (new leading term) by [tex]\(2x^2\)[/tex]:
[tex]\(-34x^4 \div 2x^2 = -17x^2\)[/tex]. Add [tex]\(-17x^2\)[/tex] to the quotient.
- Multiply and subtract again:
- Multiply: [tex]\((-17x^2)(2x^2 + x - 15) = -34x^4 - 17x^3 + 255x^2\)[/tex].
- Subtract: [tex]\((-34x^4 - 89x^3 + 201x^2) - (-34x^4 - 17x^3 + 255x^2) = -72x^3 - 54x^2\)[/tex].
5. Continue the Process:
- Divide [tex]\(-72x^3\)[/tex] by [tex]\(2x^2\)[/tex]:
[tex]\(-72x^3 \div 2x^2 = -36x\)[/tex]. Add [tex]\(-36x\)[/tex] to the quotient.
- Multiply and subtract:
- Multiply: [tex]\((-36x)(2x^2 + x - 15) = -72x^3 - 36x^2 + 540x\)[/tex].
- Subtract: [tex]\((-72x^3 - 54x^2 + 531x) - (-72x^3 - 36x^2 + 540x) = -18x^2 - 9x\)[/tex].
6. Final Steps:
- Divide [tex]\(-18x^2\)[/tex] by [tex]\(2x^2\)[/tex]:
[tex]\(-18x^2 \div 2x^2 = -9\)[/tex]. Add [tex]\(-9\)[/tex] to the quotient.
- Multiply and subtract:
- Multiply: [tex]\((-9)(2x^2 + x - 15) = -18x^2 - 9x + 135\)[/tex].
- Subtract: [tex]\((-18x^2 - 9x + 135) - (-18x^2 - 9x + 135) = 0\)[/tex].
Thus, the quotient of the division is [tex]\(10x^3 - 17x^2 - 36x - 9\)[/tex] with a remainder of [tex]\(0\)[/tex].
So, the solution to the polynomial division is:
[tex]\[
\frac{20x^5 - 24x^4 - 239x^3 + 201x^2 + 531x + 135}{2x^2 + x - 15} = 10x^3 - 17x^2 - 36x - 9
\][/tex]
