Answer :
To solve the synthetic division problem and find the remainder, we follow these steps:
1. Identify the Divisor and the Coefficients:
- The problem is structured for synthetic division with a divisor of [tex]\( x - 1 \)[/tex]. This means we will use the value [tex]\( 1 \)[/tex] as our "root" or the number for synthetic division.
- The coefficients of the polynomial (the dividend) are [tex]\( 2, -3, \)[/tex] and [tex]\( 2 \)[/tex].
2. Set Up the Synthetic Division:
- Write down the root, [tex]\( 1 \)[/tex], and place the coefficients in a row: [tex]\( 2, -3, 2 \)[/tex].
3. Perform Synthetic Division:
- Bring down the first coefficient, [tex]\( 2 \)[/tex].
- Multiply this number by the root [tex]\( 1 \)[/tex] and place the result under the next coefficient.
- Add the second coefficient, [tex]\( -3 \)[/tex], to the result from the previous step.
- Repeat these steps until you reach the last coefficient.
Let's illustrate:
- Start with the leading coefficient: [tex]\( 2 \)[/tex].
- Multiply [tex]\( 2 \)[/tex] by [tex]\( 1 \)[/tex] (the root) to get [tex]\( 2 \)[/tex].
- Add this to the next coefficient: [tex]\((-3) + 2 = -1\)[/tex].
- Multiply [tex]\(-1\)[/tex] by [tex]\( 1 \)[/tex] to get [tex]\(-1\)[/tex].
- Add this to the last coefficient: [tex]\( 2 + (-1) = 1\)[/tex].
4. Determine the Remainder:
- The last number you obtain in this process is the remainder.
So, the remainder in this synthetic division process is [tex]\( 1 \)[/tex].
Therefore, the answer is: [tex]\( \boxed{1} \)[/tex]
1. Identify the Divisor and the Coefficients:
- The problem is structured for synthetic division with a divisor of [tex]\( x - 1 \)[/tex]. This means we will use the value [tex]\( 1 \)[/tex] as our "root" or the number for synthetic division.
- The coefficients of the polynomial (the dividend) are [tex]\( 2, -3, \)[/tex] and [tex]\( 2 \)[/tex].
2. Set Up the Synthetic Division:
- Write down the root, [tex]\( 1 \)[/tex], and place the coefficients in a row: [tex]\( 2, -3, 2 \)[/tex].
3. Perform Synthetic Division:
- Bring down the first coefficient, [tex]\( 2 \)[/tex].
- Multiply this number by the root [tex]\( 1 \)[/tex] and place the result under the next coefficient.
- Add the second coefficient, [tex]\( -3 \)[/tex], to the result from the previous step.
- Repeat these steps until you reach the last coefficient.
Let's illustrate:
- Start with the leading coefficient: [tex]\( 2 \)[/tex].
- Multiply [tex]\( 2 \)[/tex] by [tex]\( 1 \)[/tex] (the root) to get [tex]\( 2 \)[/tex].
- Add this to the next coefficient: [tex]\((-3) + 2 = -1\)[/tex].
- Multiply [tex]\(-1\)[/tex] by [tex]\( 1 \)[/tex] to get [tex]\(-1\)[/tex].
- Add this to the last coefficient: [tex]\( 2 + (-1) = 1\)[/tex].
4. Determine the Remainder:
- The last number you obtain in this process is the remainder.
So, the remainder in this synthetic division process is [tex]\( 1 \)[/tex].
Therefore, the answer is: [tex]\( \boxed{1} \)[/tex]
