Question

Use synthetic division to evaluate (3x2 − 6x + 3) ÷ (x − 1).

3x − 9
3x2 − 3x
3x − 3
3x2 − 6

Answer 1

To evaluate \( (3x^2 - 6x + 3) \div (x - 1) \) using synthetic division, we'll follow these steps:

1. **Identify the coefficients** of the polynomial \( 3x^2 - 6x + 3 \). The coefficients are:
- \( 3 \) (for \( x^2 \))
- \( -6 \) (for \( x^1 \))
- \( 3 \) (for the constant term)

So we have the coefficients: \( [3, -6, 3] \).

2. **Use the root of the divisor** \( x - 1 \), which is \( 1 \).

3. **Set up the synthetic division**:

\[
\begin{array}{r|rrr}
1 & 3 & -6 & 3 \\
& & 3 & -3 \\
\hline
& 3 & -3 & 0 \\
\end{array}
\]

Here's how we perform the synthetic division:
- Bring down the \( 3 \).
- Multiply \( 1 \) (the divisor root) by \( 3 \) (the number we just brought down) to get \( 3 \), and add it to \(-6\) to get \(-3\).
- Multiply \( 1 \) by \(-3\) to get \(-3\), and add it to \( 3 \) to get \( 0 \).

4. **Interpret the result**:
The bottom row shows the coefficients of the quotient polynomial, which is \( 3x - 3 \), and the final value \( 0 \) indicates that \( (x - 1) \) is a factor of \( 3x^2 - 6x + 3 \).

Thus, the result of \( (3x^2 - 6x + 3) \div (x - 1) \) is

\[
\boxed{3x - 3}
\]