Answer :
- Bring down the first coefficient.
- Multiply the divisor by the brought-down coefficient and add it to the next coefficient.
- Repeat the process until the last column.
- The last number in the bottom row is the remainder, which is $\boxed{2}$.
### Explanation
1. Understanding the Problem
We are given a synthetic division problem and asked to find the remainder. The synthetic division is set up as follows:
$1 \longdiv { 1 \quad 2 \quad -3 \quad 2}$
This corresponds to dividing the polynomial $x^3 + 2x^2 - 3x + 2$ by $x-1$.
2. Performing Synthetic Division
To perform synthetic division, we follow these steps:
1. Bring down the first coefficient (1).
$1 \longdiv { 1 \quad 2 \quad -3 \quad 2}$
$\quad\quad\quad 1$
2. Multiply the divisor (1) by the brought-down coefficient (1) and write the result (1) under the next coefficient (2).
$1 \longdiv { 1 \quad 2 \quad -3 \quad 2}$
$\quad\quad\quad 1$
$\quad\quad 1$
3. Add the numbers in the second column (2 + 1 = 3).
$1 \longdiv { 1 \quad 2 \quad -3 \quad 2}$
$\quad\quad\quad 1$
$\quad\quad 1 \quad 3$
4. Multiply the divisor (1) by the result (3) and write the result (3) under the next coefficient (-3).
$1 \longdiv { 1 \quad 2 \quad -3 \quad 2}$
$\quad\quad\quad 1 \quad 3$
$\quad\quad 1 \quad 3$
5. Add the numbers in the third column (-3 + 3 = 0).
$1 \longdiv { 1 \quad 2 \quad -3 \quad 2}$
$\quad\quad\quad 1 \quad 3$
$\quad\quad 1 \quad 3 \quad 0$
6. Multiply the divisor (1) by the result (0) and write the result (0) under the next coefficient (2).
$1 \longdiv { 1 \quad 2 \quad -3 \quad 2}$
$\quad\quad\quad 1 \quad 3 \quad 0$
$\quad\quad 1 \quad 3 \quad 0$
7. Add the numbers in the last column (2 + 0 = 2). The result is the remainder.
$1 \longdiv { 1 \quad 2 \quad -3 \quad 2}$
$\quad\quad\quad 1 \quad 3 \quad 0$
$\quad\quad 1 \quad 3 \quad 0 \quad 2$
The remainder is 2.
3. Finding the Remainder
The remainder of the synthetic division is 2.
### Examples
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x - a. It's often used in engineering to simplify complex polynomial expressions that model physical systems, such as control systems or signal processing. For example, when designing a filter, engineers use synthetic division to analyze the transfer function of the filter, which is a ratio of two polynomials. By finding the roots and remainders, they can optimize the filter's performance to meet specific requirements.
- Multiply the divisor by the brought-down coefficient and add it to the next coefficient.
- Repeat the process until the last column.
- The last number in the bottom row is the remainder, which is $\boxed{2}$.
### Explanation
1. Understanding the Problem
We are given a synthetic division problem and asked to find the remainder. The synthetic division is set up as follows:
$1 \longdiv { 1 \quad 2 \quad -3 \quad 2}$
This corresponds to dividing the polynomial $x^3 + 2x^2 - 3x + 2$ by $x-1$.
2. Performing Synthetic Division
To perform synthetic division, we follow these steps:
1. Bring down the first coefficient (1).
$1 \longdiv { 1 \quad 2 \quad -3 \quad 2}$
$\quad\quad\quad 1$
2. Multiply the divisor (1) by the brought-down coefficient (1) and write the result (1) under the next coefficient (2).
$1 \longdiv { 1 \quad 2 \quad -3 \quad 2}$
$\quad\quad\quad 1$
$\quad\quad 1$
3. Add the numbers in the second column (2 + 1 = 3).
$1 \longdiv { 1 \quad 2 \quad -3 \quad 2}$
$\quad\quad\quad 1$
$\quad\quad 1 \quad 3$
4. Multiply the divisor (1) by the result (3) and write the result (3) under the next coefficient (-3).
$1 \longdiv { 1 \quad 2 \quad -3 \quad 2}$
$\quad\quad\quad 1 \quad 3$
$\quad\quad 1 \quad 3$
5. Add the numbers in the third column (-3 + 3 = 0).
$1 \longdiv { 1 \quad 2 \quad -3 \quad 2}$
$\quad\quad\quad 1 \quad 3$
$\quad\quad 1 \quad 3 \quad 0$
6. Multiply the divisor (1) by the result (0) and write the result (0) under the next coefficient (2).
$1 \longdiv { 1 \quad 2 \quad -3 \quad 2}$
$\quad\quad\quad 1 \quad 3 \quad 0$
$\quad\quad 1 \quad 3 \quad 0$
7. Add the numbers in the last column (2 + 0 = 2). The result is the remainder.
$1 \longdiv { 1 \quad 2 \quad -3 \quad 2}$
$\quad\quad\quad 1 \quad 3 \quad 0$
$\quad\quad 1 \quad 3 \quad 0 \quad 2$
The remainder is 2.
3. Finding the Remainder
The remainder of the synthetic division is 2.
### Examples
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x - a. It's often used in engineering to simplify complex polynomial expressions that model physical systems, such as control systems or signal processing. For example, when designing a filter, engineers use synthetic division to analyze the transfer function of the filter, which is a ratio of two polynomials. By finding the roots and remainders, they can optimize the filter's performance to meet specific requirements.
