Answer :
We are given the polynomial
[tex]$$
f(x) = 2x^3 - 4x^2 - 9x + 1
$$[/tex]
and we want to divide it by [tex]$(x-2)$[/tex] using synthetic division. Here is the step-by-step process:
1. Write down the coefficients of the polynomial:
[tex]$$2,\ -4,\ -9,\ 1$$[/tex]
The divisor value is [tex]$2$[/tex] (because [tex]$x-2=0$[/tex] gives [tex]$x=2$[/tex]).
2. Begin the process by bringing down the leading coefficient:
[tex]$$2$$[/tex]
3. Multiply this value by the divisor:
[tex]$$2 \times 2 = 4$$[/tex]
Write the [tex]$4$[/tex] under the next coefficient [tex]$-4$[/tex].
4. Add the column:
[tex]$$-4 + 4 = 0$$[/tex]
5. Multiply the result by the divisor:
[tex]$$0 \times 2 = 0$$[/tex]
Write the [tex]$0$[/tex] under the next coefficient [tex]$-9$[/tex].
6. Add the column:
[tex]$$-9 + 0 = -9$$[/tex]
7. Multiply the result by the divisor:
[tex]$$-9 \times 2 = -18$$[/tex]
Write the [tex]$-18$[/tex] under the next coefficient [tex]$1$[/tex].
8. Add the final column:
[tex]$$1 + (-18) = -17$$[/tex]
The synthetic division table looks like:
[tex]$$
\begin{array}{r|cccc}
2 & 2 & -4 & -9 & 1 \\
& & 4 & 0 & -18 \\
\hline
& 2 & 0 & -9 & -17
\end{array}
$$[/tex]
The last number, [tex]$-17$[/tex], is the remainder.
Thus, the remainder when dividing [tex]$2x^3 - 4x^2 - 9x + 1$[/tex] by [tex]$(x-2)$[/tex] is [tex]$\boxed{-17}$[/tex].
[tex]$$
f(x) = 2x^3 - 4x^2 - 9x + 1
$$[/tex]
and we want to divide it by [tex]$(x-2)$[/tex] using synthetic division. Here is the step-by-step process:
1. Write down the coefficients of the polynomial:
[tex]$$2,\ -4,\ -9,\ 1$$[/tex]
The divisor value is [tex]$2$[/tex] (because [tex]$x-2=0$[/tex] gives [tex]$x=2$[/tex]).
2. Begin the process by bringing down the leading coefficient:
[tex]$$2$$[/tex]
3. Multiply this value by the divisor:
[tex]$$2 \times 2 = 4$$[/tex]
Write the [tex]$4$[/tex] under the next coefficient [tex]$-4$[/tex].
4. Add the column:
[tex]$$-4 + 4 = 0$$[/tex]
5. Multiply the result by the divisor:
[tex]$$0 \times 2 = 0$$[/tex]
Write the [tex]$0$[/tex] under the next coefficient [tex]$-9$[/tex].
6. Add the column:
[tex]$$-9 + 0 = -9$$[/tex]
7. Multiply the result by the divisor:
[tex]$$-9 \times 2 = -18$$[/tex]
Write the [tex]$-18$[/tex] under the next coefficient [tex]$1$[/tex].
8. Add the final column:
[tex]$$1 + (-18) = -17$$[/tex]
The synthetic division table looks like:
[tex]$$
\begin{array}{r|cccc}
2 & 2 & -4 & -9 & 1 \\
& & 4 & 0 & -18 \\
\hline
& 2 & 0 & -9 & -17
\end{array}
$$[/tex]
The last number, [tex]$-17$[/tex], is the remainder.
Thus, the remainder when dividing [tex]$2x^3 - 4x^2 - 9x + 1$[/tex] by [tex]$(x-2)$[/tex] is [tex]$\boxed{-17}$[/tex].
