Answer :
To solve the synthetic division problem and find the remainder, let's go through the process step by step.
### Understanding the Problem
You're given a polynomial, and you need to divide it by a binomial. The format in the problem seems to suggest a polynomial division of the form `(x^3 + 1x^2 + -3x + 2)` divided by `(x + 2)`. This would mean:
- Dividend: [tex]\( x^3 + x^2 - 3x + 2 \)[/tex]
- Divisor: [tex]\( x + 2 \)[/tex]
### Setting Up Synthetic Division
Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form [tex]\( x - c \)[/tex]. In this problem, our divisor is [tex]\( x + 2 \)[/tex], which means [tex]\( c = -2 \)[/tex].
1. List the coefficients of the polynomial: `1` (for [tex]\( x^3 \)[/tex]), `1` (for [tex]\( x^2 \)[/tex]), `-3` (for [tex]\( x \)[/tex]), and `2` (constant term).
2. Write these coefficients in a row:
`1, 1, -3, 2`
3. Place [tex]\( c = -2 \)[/tex] to the left.
### Performing the Division
1. Bring down the leading coefficient: Start with `1`, which is the first coefficient.
2. Multiply and Add:
- Multiply `-2` by the number you just brought down (`1`) and write the result under the next coefficient (`1`).
- Add: `1 + (-2) = -1`.
- Multiply `-2` by `-1` and write the result under the next coefficient (`-3`).
- Add: `-3 + 2 = -1`.
- Multiply `-2` by `-1` and write the result under the next coefficient (`2`).
- Add: `2 + 2 = 0`.
The final row after completing synthetic division is:
`1, -1, -1, 0`
The last number, `0`, is the remainder.
### Conclusion
The remainder of dividing [tex]\( x^3 + x^2 - 3x + 2 \)[/tex] by [tex]\( x + 2 \)[/tex] using synthetic division is 0.
### Understanding the Problem
You're given a polynomial, and you need to divide it by a binomial. The format in the problem seems to suggest a polynomial division of the form `(x^3 + 1x^2 + -3x + 2)` divided by `(x + 2)`. This would mean:
- Dividend: [tex]\( x^3 + x^2 - 3x + 2 \)[/tex]
- Divisor: [tex]\( x + 2 \)[/tex]
### Setting Up Synthetic Division
Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form [tex]\( x - c \)[/tex]. In this problem, our divisor is [tex]\( x + 2 \)[/tex], which means [tex]\( c = -2 \)[/tex].
1. List the coefficients of the polynomial: `1` (for [tex]\( x^3 \)[/tex]), `1` (for [tex]\( x^2 \)[/tex]), `-3` (for [tex]\( x \)[/tex]), and `2` (constant term).
2. Write these coefficients in a row:
`1, 1, -3, 2`
3. Place [tex]\( c = -2 \)[/tex] to the left.
### Performing the Division
1. Bring down the leading coefficient: Start with `1`, which is the first coefficient.
2. Multiply and Add:
- Multiply `-2` by the number you just brought down (`1`) and write the result under the next coefficient (`1`).
- Add: `1 + (-2) = -1`.
- Multiply `-2` by `-1` and write the result under the next coefficient (`-3`).
- Add: `-3 + 2 = -1`.
- Multiply `-2` by `-1` and write the result under the next coefficient (`2`).
- Add: `2 + 2 = 0`.
The final row after completing synthetic division is:
`1, -1, -1, 0`
The last number, `0`, is the remainder.
### Conclusion
The remainder of dividing [tex]\( x^3 + x^2 - 3x + 2 \)[/tex] by [tex]\( x + 2 \)[/tex] using synthetic division is 0.
