Answer :
(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]
