Answer :
Sure! Let's find the value of the function [tex]\( f(x) = 3x^3 - 4x^2 - 5x + 4 \)[/tex] at [tex]\( x = -2 \)[/tex] using synthetic division and the Remainder Theorem.
Step-by-step Solution:
1. Set up synthetic division:
- We're evaluating [tex]\( f(-2) \)[/tex], so our divisor is [tex]\( x = -2 \)[/tex].
- Write down the coefficients of the polynomial: 3, -4, -5, and 4.
2. Perform synthetic division:
- Start by bringing down the leading coefficient (3) to the bottom row.
| | -2 | | | |
|-----|-----|-------|-----|----|
| 3 | | | | |
| | | | | |
- Multiply -2 by 3 and write the result under the next coefficient (-4).
| | -2 | | | |
|-----|-----|-------|-----|---|
| 3 | | | | |
| | -6 | | | |
- Add -6 to -4 to get -10 and write it in the new row.
| | -2 | | | |
|-----|-----|-------|-----|---|
| 3 | -10 | | | |
| | -6 | | | |
- Multiply -2 by -10 and write the result under the next coefficient (-5).
| | -2 | | | |
|-----|-----|-------|-----|----|
| 3 | -10 | | | |
| | -6 | 20 | | |
- Add 20 to -5 to get 15 and write it in the new row.
| | -2 | | | |
|-----|-----|-------|-----|---|
| 3 | -10 | 15 | | |
| | -6 | 20 | | |
- Multiply -2 by 15 and write the result under the last coefficient (4).
| | -2 | | | |
|-----|-----|-------|-----|----|
| 3 | -10 | 15 | | |
| | -6 | 20 | -30 | |
- Add -30 to 4 to get -26 and write it in the new row.
| | -2 | | | |
|-----|-----|-------|-----|-----|
| 3 | -10 | 15 | -26 | |
| | -6 | 20 | -30 | |
3. Remainder Theorem:
- The final value obtained in the last column after performing synthetic division is the remainder. In this case, it is -26.
- According to the Remainder Theorem, the value of [tex]\( f(x) \)[/tex] at [tex]\( x = -2 \)[/tex] is the remainder, which is [tex]\(-26\)[/tex].
Thus, the value of the function [tex]\( f(-2) \)[/tex] is [tex]\(-26\)[/tex].
Step-by-step Solution:
1. Set up synthetic division:
- We're evaluating [tex]\( f(-2) \)[/tex], so our divisor is [tex]\( x = -2 \)[/tex].
- Write down the coefficients of the polynomial: 3, -4, -5, and 4.
2. Perform synthetic division:
- Start by bringing down the leading coefficient (3) to the bottom row.
| | -2 | | | |
|-----|-----|-------|-----|----|
| 3 | | | | |
| | | | | |
- Multiply -2 by 3 and write the result under the next coefficient (-4).
| | -2 | | | |
|-----|-----|-------|-----|---|
| 3 | | | | |
| | -6 | | | |
- Add -6 to -4 to get -10 and write it in the new row.
| | -2 | | | |
|-----|-----|-------|-----|---|
| 3 | -10 | | | |
| | -6 | | | |
- Multiply -2 by -10 and write the result under the next coefficient (-5).
| | -2 | | | |
|-----|-----|-------|-----|----|
| 3 | -10 | | | |
| | -6 | 20 | | |
- Add 20 to -5 to get 15 and write it in the new row.
| | -2 | | | |
|-----|-----|-------|-----|---|
| 3 | -10 | 15 | | |
| | -6 | 20 | | |
- Multiply -2 by 15 and write the result under the last coefficient (4).
| | -2 | | | |
|-----|-----|-------|-----|----|
| 3 | -10 | 15 | | |
| | -6 | 20 | -30 | |
- Add -30 to 4 to get -26 and write it in the new row.
| | -2 | | | |
|-----|-----|-------|-----|-----|
| 3 | -10 | 15 | -26 | |
| | -6 | 20 | -30 | |
3. Remainder Theorem:
- The final value obtained in the last column after performing synthetic division is the remainder. In this case, it is -26.
- According to the Remainder Theorem, the value of [tex]\( f(x) \)[/tex] at [tex]\( x = -2 \)[/tex] is the remainder, which is [tex]\(-26\)[/tex].
Thus, the value of the function [tex]\( f(-2) \)[/tex] is [tex]\(-26\)[/tex].