Answer :
To solve the synthetic division problem and find the remainder, let's go through the steps:
The problem involves dividing a polynomial by [tex]\( x - 1 \)[/tex], which means that 1 is the root we will use for synthetic division. The polynomial given can be represented by the coefficients [tex]\( 1, 2, -3, 2 \)[/tex]. These correspond to the polynomial [tex]\( x^3 + 2x^2 - 3x + 2 \)[/tex].
Here's how we perform the synthetic division:
1. Write down the root:
The root for synthetic division here is 1 because the divisor is [tex]\( x - 1 \)[/tex].
2. List the coefficients:
Write the coefficients of the polynomial in a row:
[tex]\[
1 \quad 2 \quad -3 \quad 2
\][/tex]
3. Bring down the leading coefficient:
Bring down the first coefficient (1) to the bottom row.
4. Multiply and add:
- Multiply the root (1) by the value you just brought down (1) and write the result under the next coefficient:
[tex]\[
1 \times 1 = 1
\][/tex]
- Add this to the next coefficient (2):
[tex]\[
2 + 1 = 3
\][/tex]
5. Repeat for each column:
- Multiply the root (1) by the new value (3) and write the result under the next coefficient:
[tex]\[
1 \times 3 = 3
\][/tex]
- Add this to the next coefficient (-3):
[tex]\[
-3 + 3 = 0
\][/tex]
- Multiply the root (1) by this result (0) and write it under the last coefficient:
[tex]\[
1 \times 0 = 0
\][/tex]
- Add this to the final coefficient (2):
[tex]\[
2 + 0 = 2
\][/tex]
6. Obtain the remainder:
The final number (after you have added in the last column) is the remainder.
Therefore, the remainder of the division is 2.
So, the answer is:
D. 2
The problem involves dividing a polynomial by [tex]\( x - 1 \)[/tex], which means that 1 is the root we will use for synthetic division. The polynomial given can be represented by the coefficients [tex]\( 1, 2, -3, 2 \)[/tex]. These correspond to the polynomial [tex]\( x^3 + 2x^2 - 3x + 2 \)[/tex].
Here's how we perform the synthetic division:
1. Write down the root:
The root for synthetic division here is 1 because the divisor is [tex]\( x - 1 \)[/tex].
2. List the coefficients:
Write the coefficients of the polynomial in a row:
[tex]\[
1 \quad 2 \quad -3 \quad 2
\][/tex]
3. Bring down the leading coefficient:
Bring down the first coefficient (1) to the bottom row.
4. Multiply and add:
- Multiply the root (1) by the value you just brought down (1) and write the result under the next coefficient:
[tex]\[
1 \times 1 = 1
\][/tex]
- Add this to the next coefficient (2):
[tex]\[
2 + 1 = 3
\][/tex]
5. Repeat for each column:
- Multiply the root (1) by the new value (3) and write the result under the next coefficient:
[tex]\[
1 \times 3 = 3
\][/tex]
- Add this to the next coefficient (-3):
[tex]\[
-3 + 3 = 0
\][/tex]
- Multiply the root (1) by this result (0) and write it under the last coefficient:
[tex]\[
1 \times 0 = 0
\][/tex]
- Add this to the final coefficient (2):
[tex]\[
2 + 0 = 2
\][/tex]
6. Obtain the remainder:
The final number (after you have added in the last column) is the remainder.
Therefore, the remainder of the division is 2.
So, the answer is:
D. 2
