Answer :
- Perform synthetic division.
- The last number in the last row is the remainder.
- The remainder is 3.
- The final answer is $\boxed{3}$.
### Explanation
1. Understanding Synthetic Division
We are given a synthetic division problem and asked to find the remainder. The synthetic division is set up as follows:
$1
vert 1
quad 2
quad -3
quad 3$
To perform synthetic division, we bring down the first number (1), multiply it by the divisor (1), and add it to the next number (2). Then we multiply the result by the divisor (1) and add it to the next number (-3), and so on.
2. Bring Down the First Number
Step 1: Bring down the 1.
$1
vert 1
quad 2
quad -3
quad 3$
$\overline{
quad 1}$
3. Multiply and Add
Step 2: Multiply 1 by 1 and add to 2.
$1
vert 1
quad 2
quad -3
quad 3$
$\overline{
quad 1
quad 3}$
4. Multiply and Add
Step 3: Multiply 3 by 1 and add to -3.
$1
vert 1
quad 2
quad -3
quad 3$
$\overline{
quad 1
quad 3
quad 0}$
5. Multiply and Add
Step 4: Multiply 0 by 1 and add to 3.
$1
vert 1
quad 2
quad -3
quad 3$
$\overline{
quad 1
quad 3
quad 0
quad 3}$
6. Finding the Remainder
The last number in the bottom row is the remainder, which is 3.
7. Conclusion
Therefore, the remainder is 3.
### 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 algebra to find the roots of polynomials or to simplify expressions. For example, if you're designing a bridge and need to calculate the bending moment of a beam, you might use polynomial equations. Synthetic division can help simplify these calculations, making it easier to determine the structural integrity of the bridge.
- The last number in the last row is the remainder.
- The remainder is 3.
- The final answer is $\boxed{3}$.
### Explanation
1. Understanding Synthetic Division
We are given a synthetic division problem and asked to find the remainder. The synthetic division is set up as follows:
$1
vert 1
quad 2
quad -3
quad 3$
To perform synthetic division, we bring down the first number (1), multiply it by the divisor (1), and add it to the next number (2). Then we multiply the result by the divisor (1) and add it to the next number (-3), and so on.
2. Bring Down the First Number
Step 1: Bring down the 1.
$1
vert 1
quad 2
quad -3
quad 3$
$\overline{
quad 1}$
3. Multiply and Add
Step 2: Multiply 1 by 1 and add to 2.
$1
vert 1
quad 2
quad -3
quad 3$
$\overline{
quad 1
quad 3}$
4. Multiply and Add
Step 3: Multiply 3 by 1 and add to -3.
$1
vert 1
quad 2
quad -3
quad 3$
$\overline{
quad 1
quad 3
quad 0}$
5. Multiply and Add
Step 4: Multiply 0 by 1 and add to 3.
$1
vert 1
quad 2
quad -3
quad 3$
$\overline{
quad 1
quad 3
quad 0
quad 3}$
6. Finding the Remainder
The last number in the bottom row is the remainder, which is 3.
7. Conclusion
Therefore, the remainder is 3.
### 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 algebra to find the roots of polynomials or to simplify expressions. For example, if you're designing a bridge and need to calculate the bending moment of a beam, you might use polynomial equations. Synthetic division can help simplify these calculations, making it easier to determine the structural integrity of the bridge.
