Question

Which synthetic division shows how to find the quotient?

2x^3−2x^2+6x−4÷x−2

Answer 1

To perform synthetic division for the polynomial \( 2x^3 - 2x^2 + 6x - 4 \) divided by \( x - 2 \), we follow these steps:

1. **Write down the coefficients** of the polynomial:
The coefficients for \( 2x^3 - 2x^2 + 6x - 4 \) are:
\[
2, -2, 6, -4
\]

2. **Set the divisor equal to zero** to find the number we're going to use in synthetic division. For \( x - 2 \), setting it equal to zero gives:
\[
x - 2 = 0 \implies x = 2
\]

3. **Set up synthetic division**: We write \( 2 \) on the left and the coefficients on the right.
\[
\begin{array}{r|rrrr}
2 & 2 & -2 & 6 & -4 \\
& & & & \\
\end{array}
\]

4. **Bring down the leading coefficient**:
\[
\begin{array}{r|rrrr}
2 & 2 & -2 & 6 & -4 \\
& & & & \\
& 2 & & & \\
\end{array}
\]

5. **Multiply and add**:
- Multiply \( 2 \) (the number we brought down) by \( 2 \) (the divisor root) and place it under the next coefficient:
\[
\begin{array}{r|rrrr}
2 & 2 & -2 & 6 & -4 \\
& & 4 & & \\
& 2 & & & \\
\end{array}
\]
- Now, add \( -2 + 4 = 2 \):
\[
\begin{array}{r|rrrr}
2 & 2 & -2 & 6 & -4 \\
& & 4 & & \\
& 2 & 2 & & \\
\end{array}
\]

6. **Continue the process**:
- Multiply \( 2 \) (the result just computed) by \( 2 \):
\[
\begin{array}{r|rrrr}
2 & 2 & -2 & 6 & -4 \\
& & 4 & 4 & \\
& 2 & 2 & & \\
\end{array}
\]
- Add \( 6 + 4 = 10 \):
\[
\begin{array}{r|rrrr}
2 & 2 & -2 & 6 & -4 \\
& & 4 & 4 & \\
& 2 & 2 & 10 & \\
\end{array}
\]

7. **Final multiplication and addition**:
- Multiply \( 10 \) by \( 2 \):
\[
\begin{array}{r|rrrr}
2 & 2 & -2 & 6 & -4 \\
& & 4 & 4 & 20 \\
& 2 & 2 & 10 & \\
\end{array}
\]
- Add \( -4 + 20 = 16 \):
\[
\begin{array}{r|rrrr}
2 & 2 & -2 & 6 & -4 \\
& & 4 & 4 & 20 \\
& 2 & 2 & 10 & 16 \\
\end{array}
\]

8. **Write the result**: The bottom row (excluding the last number, which is the remainder) forms the coefficients of the quotient polynomial. Here, the quotient is:
\[
2x^2 + 2x + 10
\]
And the remainder is \( 16 \).

So, the final result of the synthetic division is:
\[
\frac{2x^3 - 2x^2 + 6x - 4}{x - 2} = 2x^2 + 2x + 10 \quad \text{with a remainder of } 16.
\]