To divide the polynomials using synthetic division, we need to follow these steps:
1. Rewrite the divisor: The given divisor is [tex]\( -\frac{1}{4}w + 2 \)[/tex]. To use synthetic division, we need to represent it in the form [tex]\( (w - r) \)[/tex]. By multiplying the divisor by [tex]\(-4\)[/tex], we get [tex]\( w - 8 \)[/tex]. Therefore, [tex]\( r = 8 \)[/tex].
2. Identify the coefficients and terms:
- The coefficients of the dividend polynomial [tex]\( - \frac { 5 } { 4 } w ^ { 5 } + 9 w ^ { 4 } + \frac { 2 9 } { 4 } w ^ { 3 } + \frac { 1 3 } { 2 } w ^ { 2 } - \frac { 1 1 } { 4 } w - 1 3 \)[/tex] are:
[tex]\[ \left[ -\frac{5}{4}, 9, \frac{29}{4}, \frac{13}{2}, -\frac{11}{4}, -13 \right] \][/tex]
3. Set up synthetic division:
- Write the root [tex]\( r \)[/tex] on the left, and the coefficients of the dividend polynomial on the top row:
[tex]\[
\begin{array}{c|cccccc}
8 & -\frac{5}{4} & 9 & \frac{29}{4} & \frac{13}{2} & -\frac{11}{4} & -13 \\
\end{array}
\][/tex]
4. Perform synthetic division step-by-step:
- Start with the first coefficient: [tex]\( -\frac{5}{4} \)[/tex].
- Multiply this by [tex]\( 8 \)[/tex] and add it to the next coefficient [tex]\( 9 \)[/tex]:
[tex]\[
\begin{array}{c|cccccc}
8 & -\frac{5}{4} & 9 & \frac{29}{4} & \frac{13}{2} & -\frac{11}{4} & -13 \\
& & -10 & & & & \\
\hline
& & -1 & & & & \\
\end{array}
\][/tex]
- Continue this process for each coefficient:
[tex]\[
\begin{array}{c|cccccc}
8 & -\frac{5}{4} & 9 & \frac{29}{4} & \frac{13}{2} & -\frac{11}{4} & -13 \\
& & -10 & 79 & & & \\
& \cdot & -79 & 25 & & & \\
\hline
& -\frac{5}{4} & 19 & -144.75 & 1164.5 & -9318.75 & 0 \\
\end{array}
\][/tex]
and include the final value at the end to calculate the remainder:
[tex]\[
\begin{array}{c|cccccc}
8 & -\frac{5}{4} & 9 & \frac{29}{4} & \frac{13}{2} & -\frac{11}{4} & -13 \\
& & -10 & 79 & -625 & 4661 & -37338 \\
& -\frac{5}{4} & 19 & -144.75 & 1164.5 & -9318.75 & 74537 \\
\end{array}
\][/tex]
The result of synthetic division is:
- Quotient: [tex]\[-1.25, 19.0, -144.75, 1164.5, -9318.75\][/tex]
- Remainder: [tex]\( 74537.0 \)[/tex]
Answer the specific questions:
- Is the divisor given in [tex]\((x-r)\)[/tex] form?
[tex]\[
\boxed{\text{True}}
\][/tex]
- How many terms are in the dividend polynomial?
[tex]\[
\boxed{6}
\][/tex]
- Enter the quotient and remainder:
[tex]\[
\text{Quotient: } \boxed{[-1.25, 19.0, -144.75, 1164.5, -9318.75]}
\][/tex]
[tex]\[
\text{Remainder: } \boxed{74537.0}
\][/tex]
