Sure, let's solve [tex]\(\frac{x^5 + 3x^3 - 4x^2 + 2x + 6}{x-1}\)[/tex] using synthetic division.
To use synthetic division when dividing by [tex]\(x - c\)[/tex] (here [tex]\(c = 1\)[/tex]), we'll follow these steps:
1. Write down the coefficients of the dividend [tex]\(x^5 + 3x^3 - 4x^2 + 2x + 6\)[/tex]:
[tex]\( [1, 0, 3, -4, 2, 6] \)[/tex]
2. Set up the synthetic division process:
We will have a row for coefficients and then the calculations underneath:
[tex]\[
\begin{array}{r|rrrrrr}
1 & 1 & 0 & 3 & -4 & 2 & 6 \\
\hline
& & & & & & \\
\end{array}
\][/tex]
3. Drop the first coefficient down as it is:
[tex]\[
\begin{array}{r|rrrrrr}
1 & 1 & 0 & 3 & -4 & 2 & 6 \\
\hline
& 1 & & & & & \\
\end{array}
\][/tex]
4. Multiply this number by [tex]\(c = 1\)[/tex] and write the result under the next coefficient:
[tex]\[
\begin{array}{r|rrrrrr}
1 & 1 & 0 & 3 & -4 & 2 & 6 \\
\hline
& 1 & 1 & & & & \\
\end{array}
\][/tex]
5. Add the numbers in the current column:
[tex]\[
\begin{array}{r|rrrrrr}
1 & 1 & 0 & 3 & -4 & 2 & 6 \\
\hline
& 1 & 1 & 4 & & & \\
\end{array}
\][/tex]
6. Repeat the multiplication and addition steps for the remaining coefficients:
Multiply the last result by [tex]\(c = 1\)[/tex] and write it under the next coefficient:
[tex]\[
\begin{array}{r|rrrrrr}
1 & 1 & 0 & 3 & -4 & 2 & 6 \\
\hline
& 1 & 1 & 4 & 0 & 2 & \\
\end{array}
\][/tex]
Keep repeating the steps:
[tex]\[
\begin{array}{r|rrrrrr}
1 & 1 & 0 & 3 & -4 & 2 & 6 \\
\hline
& 1 & 1 & 4 & 0 & 2 & 8 \\
\end{array}
\][/tex]
7. The final result:
The numbers on the bottom row (except the last number which is the remainder) are the coefficients of the quotient polynomial. So, the quotient is:
[tex]\[
x^4 + x^3 + 4x^2 + 0x + 2 = x^4 + x^3 + 4x^2 + 2
\][/tex]
The last number [tex]\(8\)[/tex] is the remainder.
Therefore, the quotient and remainder are:
[tex]\[
\boxed{x^4 + x^3 + 4x^2 + 2 \quad \text{with a remainder of} \quad 8}
\][/tex]
