To solve the synthetic division of [tex]\( \frac{3x^2 - 4x + 9}{x - 2} \)[/tex], we follow a step-by-step process.
1. Identify the coefficients from the polynomial [tex]\( 3x^2 - 4x + 9 \)[/tex]:
- The coefficients are [tex]\( 3, -4, \)[/tex] and [tex]\( 9 \)[/tex].
2. Identify the root from the divisor [tex]\( x - 2 \)[/tex]:
- The root (value of [tex]\( x \)[/tex]) is [tex]\( 2\)[/tex].
3. Set up the synthetic division:
- Write down the root ([tex]\( 2 \)[/tex]) on the left.
- List the coefficients ([tex]\( 3, -4, 9 \)[/tex]) on the right, typically separated by a bar.
The setup will look like this:
[tex]\[
2 \mid \begin{array}{ccc}
3 & -4 & 9 \\
\end{array}
\][/tex]
4. Begin the synthetic division process:
- Bring down the first coefficient (3) as it is.
[tex]\[
2 \mid \begin{array}{ccc}
3 & -4 & 9 \\
\downarrow && \\
& 3 &
\end{array}
\][/tex]
5. Multiply the root (2) by the value just written below the bar (3) and place the result below the next coefficient (-4):
[tex]\[
2 \times 3 = 6
\][/tex]
[tex]\[
2 \mid \begin{array}{ccc}
3 & -4 & 9 \\
\downarrow & 6 & \\
& 3 & \\
\end{array}
\][/tex]
6. Add this result (6) to the next coefficient (-4):
[tex]\[
-4 + 6 = 2
\][/tex]
Write 2 below the bar:
[tex]\[
2 \mid \begin{array}{ccc}
3 & -4 & 9 \\
\downarrow & 6 & \\
& 3 & 2 \\
\end{array}
\][/tex]
7. Multiply the root (2) by the new value (2) written below the bar and place the result below the next coefficient (9):
[tex]\[
2 \times 2 = 4
\][/tex]
[tex]\[
2 \mid \begin{array}{ccc}
3 & -4 & 9 \\
\downarrow & 6 & 4 \\
& 3 & 2 \\
\end{array}
\][/tex]
8. Add this result (4) to the next coefficient (9):
[tex]\[
9 + 4 = 13
\][/tex]
Write 13 below the bar:
[tex]\[
2 \mid \begin{array}{ccc}
3 & -4 & 9 \\
\downarrow & 6 & 4 \\
& 3 & 2 & 13 \\
\end{array}
\][/tex]
The final row [tex]\( 3, 2, 13 \)[/tex] represents the coefficients of the quotient polynomial and the remainder. The quotient is [tex]\( 3x + 2 \)[/tex] and the remainder is 13.
Therefore, the synthetic division of [tex]\( \frac{3x^2 - 4x + 9}{x - 2} \)[/tex] yields:
[tex]\[
3x + 2 \text{ with a remainder of } 13
\][/tex]
Based on the provided options, the correct synthetic division form is:
C. [tex]\( 2 \longdiv { 3 \quad -4 \quad 9 } \)[/tex]
