Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Discover a wealth of knowledge from experts across different disciplines on our comprehensive Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
(a) Let's complete the synthetic division table for dividing [tex]\(-5x^5 + 0x^4 + 17x^3 + 0x^2 - x + 8\)[/tex] by [tex]\(x-3\)[/tex]:
1. Write down the coefficients of the polynomial: [tex]\([-5, 0, 17, 0, -1, 8]\)[/tex].
2. Set up the synthetic division table with [tex]\(3\)[/tex] as the divisor:
[tex]\[ \begin{array}{ccccccc} 3 & & -5 & 0 & 17 & 0 & -1 & 8 \\ & & & \downarrow& \downarrow & \downarrow & \downarrow & \downarrow \\ & & -5 & -15 & -28 & -84 & -253 & -751 \\ \end{array} \][/tex]
3. Perform the synthetic division:
- Start by bringing down [tex]\(-5\)[/tex], the leading coefficient:
[tex]\[ \begin{array}{ccccccc} 3 & & -5 & & & & & \\ & & -5 & & & & & \\ \end{array} \][/tex]
- Multiply [tex]\(-5\)[/tex] by [tex]\(3\)[/tex] and write the result under the next coefficient:
[tex]\[ \begin{array}{ccccccc} 3 & & -5 & 0 & & & & \\ & & -5 & -15 & & & & \\ \end{array} \][/tex]
- Add [tex]\(0\)[/tex] and [tex]\(-15\)[/tex]:
[tex]\[ \begin{array}{ccccccc} 3 & & -5 & 0 & 17 & & & \\ & & -5 & -15 & & & & \\ & & -5 & -15 & 2 & & & \\ \end{array} \][/tex]
- Multiply [tex]\(-15\)[/tex] by [tex]\(3\)[/tex] and add to the next coefficient [tex]\(17\)[/tex]:
[tex]\[ \begin{array}{ccccccc} 3 & & -5 & 0 & 17 & 0 & & \\ & & -5 & -15 & -28 & & & \\ & & -5 & -15 & -28 & -84 & & \\ \end{array} \][/tex]
- Multiply [tex]\(-28\)[/tex] by [tex]\(3\)[/tex] and add to the next coefficient [tex]\(0\)[/tex]:
[tex]\[ \begin{array}{ccccccc} 3 & & -5 & 0 & 17 & 0 & -1 & 8 \\ & & -5 & -15 & -28 & -84 & & \\ & & -5 & -15 & -28 & -84 & -1 & \\ \end{array} \][/tex]
- Multiply [tex]\(-84\)[/tex] by [tex]\(3\)[/tex] and add to the next coefficient [tex]\(-1\)[/tex]:
[tex]\[ \begin{array}{ccccccc} 3 & & -5 & 0 & 17 & 0 & -1 & 8 \\ & & -5 & -15 & -28 & -84 & -253 & \\ & & -5 & -15 & -28 & -84 & -253 & 8 \\ \end{array} \][/tex]
- Finally, multiply [tex]\(-253\)[/tex] by [tex]\(3\)[/tex] and add to the last coefficient [tex]\(8\)[/tex]:
[tex]\[ \begin{array}{ccccccc} 3 & & -5 & 0 & 17 & 0 & -1 & 8 \\ & & -5 & -15 & -28 & -84 & -253 & -751 \\ & & -5 & -15 & -28 & -84 & -253 & -751 \\ \end{array} \][/tex]
4. The last row from left to right contains our quotient and remainder.
- Quotient: [tex]\(-5x^4 - 15x^3 - 28x^2 - 84x - 253\)[/tex]
- Remainder: [tex]\(-751\)[/tex]
(b) Writing the final result in the given form:
[tex]\[ \frac{-5x^5 + 0x^4 + 17x^3 + 0x^2 - 1x + 8}{x-3} = -5x^4 - 15x^3 - 28x^2 - 84x - 253 + \frac{-751}{x-3} \][/tex]
1. Write down the coefficients of the polynomial: [tex]\([-5, 0, 17, 0, -1, 8]\)[/tex].
2. Set up the synthetic division table with [tex]\(3\)[/tex] as the divisor:
[tex]\[ \begin{array}{ccccccc} 3 & & -5 & 0 & 17 & 0 & -1 & 8 \\ & & & \downarrow& \downarrow & \downarrow & \downarrow & \downarrow \\ & & -5 & -15 & -28 & -84 & -253 & -751 \\ \end{array} \][/tex]
3. Perform the synthetic division:
- Start by bringing down [tex]\(-5\)[/tex], the leading coefficient:
[tex]\[ \begin{array}{ccccccc} 3 & & -5 & & & & & \\ & & -5 & & & & & \\ \end{array} \][/tex]
- Multiply [tex]\(-5\)[/tex] by [tex]\(3\)[/tex] and write the result under the next coefficient:
[tex]\[ \begin{array}{ccccccc} 3 & & -5 & 0 & & & & \\ & & -5 & -15 & & & & \\ \end{array} \][/tex]
- Add [tex]\(0\)[/tex] and [tex]\(-15\)[/tex]:
[tex]\[ \begin{array}{ccccccc} 3 & & -5 & 0 & 17 & & & \\ & & -5 & -15 & & & & \\ & & -5 & -15 & 2 & & & \\ \end{array} \][/tex]
- Multiply [tex]\(-15\)[/tex] by [tex]\(3\)[/tex] and add to the next coefficient [tex]\(17\)[/tex]:
[tex]\[ \begin{array}{ccccccc} 3 & & -5 & 0 & 17 & 0 & & \\ & & -5 & -15 & -28 & & & \\ & & -5 & -15 & -28 & -84 & & \\ \end{array} \][/tex]
- Multiply [tex]\(-28\)[/tex] by [tex]\(3\)[/tex] and add to the next coefficient [tex]\(0\)[/tex]:
[tex]\[ \begin{array}{ccccccc} 3 & & -5 & 0 & 17 & 0 & -1 & 8 \\ & & -5 & -15 & -28 & -84 & & \\ & & -5 & -15 & -28 & -84 & -1 & \\ \end{array} \][/tex]
- Multiply [tex]\(-84\)[/tex] by [tex]\(3\)[/tex] and add to the next coefficient [tex]\(-1\)[/tex]:
[tex]\[ \begin{array}{ccccccc} 3 & & -5 & 0 & 17 & 0 & -1 & 8 \\ & & -5 & -15 & -28 & -84 & -253 & \\ & & -5 & -15 & -28 & -84 & -253 & 8 \\ \end{array} \][/tex]
- Finally, multiply [tex]\(-253\)[/tex] by [tex]\(3\)[/tex] and add to the last coefficient [tex]\(8\)[/tex]:
[tex]\[ \begin{array}{ccccccc} 3 & & -5 & 0 & 17 & 0 & -1 & 8 \\ & & -5 & -15 & -28 & -84 & -253 & -751 \\ & & -5 & -15 & -28 & -84 & -253 & -751 \\ \end{array} \][/tex]
4. The last row from left to right contains our quotient and remainder.
- Quotient: [tex]\(-5x^4 - 15x^3 - 28x^2 - 84x - 253\)[/tex]
- Remainder: [tex]\(-751\)[/tex]
(b) Writing the final result in the given form:
[tex]\[ \frac{-5x^5 + 0x^4 + 17x^3 + 0x^2 - 1x + 8}{x-3} = -5x^4 - 15x^3 - 28x^2 - 84x - 253 + \frac{-751}{x-3} \][/tex]
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
