Answer :
To find the remainder using synthetic division, we need to:
1. Identify the divisor and the coefficients:
- The expression given (1 longdiv { 1, 2, -3, 3 }) implies that we're dividing a polynomial by [tex]\( x - 1 \)[/tex].
- This means the root (or value for synthetic division) is [tex]\( 1 \)[/tex].
- The polynomial coefficients are [tex]\( [1, 2, -3, 3] \)[/tex].
2. Set up for synthetic division:
- Write the coefficients in a row: [tex]\( 1, 2, -3, 3 \)[/tex].
3. Perform synthetic division:
- Bring down the first coefficient, [tex]\( 1 \)[/tex].
- Multiply this [tex]\( 1 \)[/tex] by the root [tex]\( 1 \)[/tex], and write the result under the next coefficient.
- Add [tex]\( 2 \)[/tex] and [tex]\( 1 \)[/tex] to get [tex]\( 3 \)[/tex].
- Multiply [tex]\( 3 \)[/tex] by the root [tex]\( 1 \)[/tex], place it under the next coefficient, [tex]\(-3\)[/tex].
- Add [tex]\(-3\)[/tex] and [tex]\(3\)[/tex] to get [tex]\(0\)[/tex].
- Multiply [tex]\(0\)[/tex] by the root [tex]\(1\)[/tex], place it under the next coefficient, [tex]\(3\)[/tex].
- Add [tex]\(3\)[/tex] and [tex]\(0\)[/tex] to get [tex]\(3\)[/tex].
The synthetic division steps yield:
- New row (quotient and remainder): [tex]\( [1, 3, 0, 3] \)[/tex]
4. Identify the remainder:
- The last number obtained, [tex]\(3\)[/tex], is the remainder.
Therefore, the remainder of the division is [tex]\( \boxed{3} \)[/tex].
1. Identify the divisor and the coefficients:
- The expression given (1 longdiv { 1, 2, -3, 3 }) implies that we're dividing a polynomial by [tex]\( x - 1 \)[/tex].
- This means the root (or value for synthetic division) is [tex]\( 1 \)[/tex].
- The polynomial coefficients are [tex]\( [1, 2, -3, 3] \)[/tex].
2. Set up for synthetic division:
- Write the coefficients in a row: [tex]\( 1, 2, -3, 3 \)[/tex].
3. Perform synthetic division:
- Bring down the first coefficient, [tex]\( 1 \)[/tex].
- Multiply this [tex]\( 1 \)[/tex] by the root [tex]\( 1 \)[/tex], and write the result under the next coefficient.
- Add [tex]\( 2 \)[/tex] and [tex]\( 1 \)[/tex] to get [tex]\( 3 \)[/tex].
- Multiply [tex]\( 3 \)[/tex] by the root [tex]\( 1 \)[/tex], place it under the next coefficient, [tex]\(-3\)[/tex].
- Add [tex]\(-3\)[/tex] and [tex]\(3\)[/tex] to get [tex]\(0\)[/tex].
- Multiply [tex]\(0\)[/tex] by the root [tex]\(1\)[/tex], place it under the next coefficient, [tex]\(3\)[/tex].
- Add [tex]\(3\)[/tex] and [tex]\(0\)[/tex] to get [tex]\(3\)[/tex].
The synthetic division steps yield:
- New row (quotient and remainder): [tex]\( [1, 3, 0, 3] \)[/tex]
4. Identify the remainder:
- The last number obtained, [tex]\(3\)[/tex], is the remainder.
Therefore, the remainder of the division is [tex]\( \boxed{3} \)[/tex].
