To determine whether [tex]\( 4i \)[/tex] is a zero of the polynomial function [tex]\( g(x) = x^3 - 2x^2 + 4x - 8 \)[/tex], we will use synthetic division.
### Step-by-Step Synthetic Division:
1. Write down the coefficients of the polynomial:
[tex]\[
1, -2, 4, -8
\][/tex]
2. Set up the synthetic division:
We will be using [tex]\( 4i \)[/tex] as the number to test if it is a zero of the polynomial. In the synthetic division setup, we'll place [tex]\( 4i \)[/tex] on the left and the coefficients on the top row.
[tex]\[
\begin{array}{r|rrrr}
4i & 1 & -2 & 4 & -8 \\
\end{array}
\][/tex]
3. Bring down the leading coefficient:
[tex]\[
\begin{array}{r|rrrr}
4i & 1 & -2 & 4 & -8 \\
& & & & \\
& 1 & & & \\
\end{array}
\][/tex]
4. Multiply and add down the columns:
- Multiply [tex]\( 1 \)[/tex] by [tex]\( 4i \)[/tex] and add to the next coefficient:
[tex]\[
(1)(4i) = 4i
\][/tex]
Add to [tex]\(-2\)[/tex]:
[tex]\[
-2 + 4i = -2 + 4i
\][/tex]
[tex]\[
\begin{array}{r|rrrr}
4i & 1 & -2 & 4 & -8 \\
& & 4i & & \\
& 1 & -2+4i & & \\
\end{array}
\][/tex]
- Multiply [tex]\(-2 + 4i\)[/tex] by [tex]\( 4i \)[/tex] and add to the next coefficient:
[tex]\[
(-2 + 4i)(4i) = -8i -16 = -16 - 8i
\][/tex]
Add to [tex]\( 4 \)[/tex]:
[tex]\[
4 + (-16 - 8i) = -12 - 8i
\][/tex]
[tex]\[
\begin{array}{r|rrrr}
4i & 1 & -2 & 4 & -8 \\
& & 4i & -16 & -8i \\
& 1 & -2+4i & -12-8i & \\
\end{array}
\][/tex]
- Multiply [tex]\(-12 - 8i\)[/tex] by [tex]\( 4i \)[/tex] and add to the last coefficient:
[tex]\[
(-12 - 8i)(4i) = -48i + 32 = 32 - 48i
\][/tex]
Add to [tex]\(-8\)[/tex]:
[tex]\[
-8 + (32 - 48i) = 24 - 48i
\][/tex]
[tex]\[
\begin{array}{r|rrrr}
4i & 1 & -2 & 4 & -8 \\
& & 4i & -16 & -8i \\
& 1 & -2+4i & -12-8i & 24-48i \\
\end{array}
\][/tex]
5. Determine the remainder:
The final value, [tex]\( 24 - 48i \)[/tex], is the remainder of the synthetic division.
6. Conclusion:
Since the remainder is [tex]\( 24 - 48i \)[/tex], which is not zero, [tex]\( 4i \)[/tex] is not a zero of the polynomial [tex]\( g(x) = x^3 - 2x^2 + 4x - 8 \)[/tex].
