Sure! Let's perform synthetic division to find the quotient and remainder of the polynomial division [tex]\(\frac{2x^4 + x - 6}{x - 1}\)[/tex].
1. Setup: Write down the coefficients of the polynomial [tex]\(2x^4 + 0x^3 + 0x^2 + x - 6\)[/tex]. The coefficients are [tex]\(2, 0, 0, 1, -6\)[/tex].
2. Division Parameter: For the divisor [tex]\(x - 1\)[/tex], the parameter to use in synthetic division is [tex]\(1\)[/tex].
3. Synthetic Division Table Setup: Initialize with the divisor parameter ([tex]\(1\)[/tex]) and the polynomial coefficients:
- Coefficients: [tex]\( [2, 0, 0, 1, -6] \)[/tex]
4. Performing Synthetic Division:
- Step 1: Bring down the leading coefficient [tex]\(2\)[/tex].
[tex]\[
\begin{array}{c|ccccc}
1 & 2 & 0 & 0 & 1 & -6 \\
\hline
& 2 & & & & \\
\end{array}
\][/tex]
- Step 2: Multiply the divisor parameter by the value just written below the line ([tex]\(2 \cdot 1 = 2\)[/tex]) and write it under the next coefficient.
[tex]\[
\begin{array}{c|ccccc}
1 & 2 & 0 & 0 & 1 & -6 \\
\hline
& 2 & 2 & & & \\
\end{array}
\][/tex]
- Step 3: Add the value just written to the next coefficient ([tex]\(0 + 2 = 2\)[/tex]).
[tex]\[
\begin{array}{c|ccccc}
1 & 2 & 0 & 0 & 1 & -6 \\
\hline
& 2 & 2 & & & \\
\end{array}
\][/tex]
- Step 4: Repeat this process for each coefficient:
- Multiply [tex]\(2 \cdot 1 = 2\)[/tex] and add to the next coefficient [tex]\(0\)[/tex]:
[tex]\[
\begin{array}{c|ccccc}
1 & 2 & 0 & 0 & 1 & -6 \\
\hline
& 2 & 2 & 2 & & \\
\end{array}
\][/tex]
- Multiply [tex]\(2 \cdot 1 = 2\)[/tex] and add to the next coefficient [tex]\(1\)[/tex]:
[tex]\[
\begin{array}{c|ccccc}
1 & 2 & 0 & 0 & 1 & -6 \\
\hline
& 2 & 2 & 2 & 3 & \\
\end{array}
\][/tex]
- Multiply [tex]\(3 \cdot 1 = 3\)[/tex] and add to the next coefficient [tex]\(-6\)[/tex]:
[tex]\[
\begin{array}{c|ccccc}
1 & 2 & 0 & 0 & 1 & -6 \\
\hline
& 2 & 2 & 2 & 3 & -3 \\
\end{array}
\][/tex]
5. Result Interpretation:
- The bottom row except the last value [tex]\( [2, 2, 2, 3] \)[/tex] represents the coefficients of the quotient polynomial.
- The last value [tex]\(-3\)[/tex] is the remainder.
Therefore, the quotient of [tex]\(\frac{2x^4 + x - 6}{x - 1}\)[/tex] is [tex]\(2x^3 + 2x^2 + 2x + 3\)[/tex] and the remainder is [tex]\(-3\)[/tex].
The division is expressed as:
[tex]\[
\frac{2x^4 + x - 6}{x - 1} = 2x^3 + 2x^2 + 2x + 3 + \frac{-3}{x - 1}
\][/tex]
