Divide $4x^2 - 8x + 3$ by $x - 1$ using synthetic division.

Steps to Solve

1. Identify the coefficients and the divisor root

The polynomial is $4x^2 - 8x + 3$, so the coefficients are $4$, $-8$, and $3$. The divisor is $x - 1$, so the root is $1$.

2. Set up the synthetic division

Write the root ($1$) to the left and the coefficients ($4$, $-8$, $3$) to the right.

3. Bring down the first coefficient

Bring down the first coefficient ($4$) to the bottom row.

4. Multiply and add

Multiply the root ($1$) by the number you just brought down ($4$), which gives $4$. Write this under the next coefficient ($-8$). Add $-8$ and $4$ to get $-4$.

5. Repeat the process

Multiply the root ($1$) by the result ($-4$), which gives $-4$. Write this under the next coefficient ($3$). Add $3$ and $-4$ to get $-1$.

6. Interpret the result

The numbers on the bottom row are the coefficients of the quotient and the remainder. Since we started with a quadratic ($x^2$) and divided by a linear term ($x$), the quotient will be linear ($x$). The last number is the remainder. In this case, the quotient is $4x - 4$ and the remainder is $-1$.

7. Write the final answer

The result of the division is the quotient plus the remainder divided by the divisor: $4x - 4 + \frac{-1}{x-1}$.