Answer :
Sure! Let's use synthetic division to find [tex]\( f(4) \)[/tex] for the polynomial [tex]\( f(x) = 3x^4 - 14x^3 + 3x^2 + 18x + 23 \)[/tex].
Step-by-step Solution:
1. Set up the synthetic division:
- Write down the coefficients of the polynomial: [tex]\( 3, -14, 3, 18, 23 \)[/tex].
- We will synthetically divide by [tex]\( x - 4 \)[/tex], so we'll use 4 in the synthetic division process.
2. Begin the process:
- Bring down the leading coefficient, which is [tex]\( 3 \)[/tex].
3. Multiply and add:
- Multiply the number you just brought down (3) by 4 (the value you divide by) and write it under the next coefficient.
- Add the result to the next coefficient:
[tex]\[
\begin{align*}
4 \times 3 &= 12, \\
-14 + 12 &= -2.
\end{align*}
\][/tex]
- Write [tex]\(-2\)[/tex] as the next number in the row below.
4. Continue the process:
- Multiply [tex]\(-2\)[/tex] by 4 and add to the next coefficient:
[tex]\[
\begin{align*}
4 \times -2 &= -8, \\
3 + (-8) &= -5.
\end{align*}
\][/tex]
- Write [tex]\(-5\)[/tex] next.
5. Repeat the process:
- Multiply [tex]\(-5\)[/tex] by 4, add to the next coefficient:
[tex]\[
\begin{align*}
4 \times -5 &= -20, \\
18 + (-20) &= -2.
\end{align*}
\][/tex]
- Write [tex]\(-2\)[/tex] next.
6. Final step:
- Multiply [tex]\(-2\)[/tex] by 4, add to the last coefficient:
[tex]\[
\begin{align*}
4 \times -2 &= -8, \\
23 + (-8) &= 15.
\end{align*}
\][/tex]
The final result from synthetic division, which appears at the end of our calculations, is [tex]\( 15 \)[/tex].
Therefore, [tex]\( f(4) = 15 \)[/tex].
Step-by-step Solution:
1. Set up the synthetic division:
- Write down the coefficients of the polynomial: [tex]\( 3, -14, 3, 18, 23 \)[/tex].
- We will synthetically divide by [tex]\( x - 4 \)[/tex], so we'll use 4 in the synthetic division process.
2. Begin the process:
- Bring down the leading coefficient, which is [tex]\( 3 \)[/tex].
3. Multiply and add:
- Multiply the number you just brought down (3) by 4 (the value you divide by) and write it under the next coefficient.
- Add the result to the next coefficient:
[tex]\[
\begin{align*}
4 \times 3 &= 12, \\
-14 + 12 &= -2.
\end{align*}
\][/tex]
- Write [tex]\(-2\)[/tex] as the next number in the row below.
4. Continue the process:
- Multiply [tex]\(-2\)[/tex] by 4 and add to the next coefficient:
[tex]\[
\begin{align*}
4 \times -2 &= -8, \\
3 + (-8) &= -5.
\end{align*}
\][/tex]
- Write [tex]\(-5\)[/tex] next.
5. Repeat the process:
- Multiply [tex]\(-5\)[/tex] by 4, add to the next coefficient:
[tex]\[
\begin{align*}
4 \times -5 &= -20, \\
18 + (-20) &= -2.
\end{align*}
\][/tex]
- Write [tex]\(-2\)[/tex] next.
6. Final step:
- Multiply [tex]\(-2\)[/tex] by 4, add to the last coefficient:
[tex]\[
\begin{align*}
4 \times -2 &= -8, \\
23 + (-8) &= 15.
\end{align*}
\][/tex]
The final result from synthetic division, which appears at the end of our calculations, is [tex]\( 15 \)[/tex].
Therefore, [tex]\( f(4) = 15 \)[/tex].
