Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Get quick and reliable solutions to your questions from a community of seasoned experts on our user-friendly platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
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]
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.