Answer :
To solve the synthetic division problem and find the remainder, let's follow these steps:
1. Identify the polynomial and divisor: We are dividing the polynomial [tex]\(x^3 + 2x^2 - 3x + 3\)[/tex] by [tex]\(x - 1\)[/tex]. In synthetic division, we use the root of the divisor, which in this case is [tex]\(1\)[/tex] (from [tex]\(x - 1 = 0\)[/tex]).
2. Set up the synthetic division process:
- Write down the coefficients of the polynomial: [tex]\(1, 2, -3, 3\)[/tex].
- Set aside the constant or root of the divisor, which is [tex]\(1\)[/tex].
3. Perform the synthetic division:
- Start with the leading coefficient, which is [tex]\(1\)[/tex]. Bring it straight down, as this is our initial result.
- Step 1: Multiply this result ([tex]\(1\)[/tex]) by the root of the divisor ([tex]\(1\)[/tex]) and write the result under the next coefficient. Add this result to the next coefficient:
[tex]\[
2 + (1 \times 1) = 2 + 1 = 3
\][/tex]
So, replace the second coefficient with [tex]\(3\)[/tex].
- Step 2: Multiply the result from Step 1 ([tex]\(3\)[/tex]) by the root of the divisor ([tex]\(1\)[/tex]):
[tex]\[
-3 + (3 \times 1) = -3 + 3 = 0
\][/tex]
So, replace the third coefficient with [tex]\(0\)[/tex].
- Step 3: Multiply the result from Step 2 ([tex]\(0\)[/tex]) by the root of the divisor ([tex]\(1\)[/tex]):
[tex]\[
3 + (0 \times 1) = 3 + 0 = 3
\][/tex]
So, replace the last coefficient with [tex]\(3\)[/tex].
4. Determine the remainder: The last number obtained after processing all coefficients is the remainder of the division. In this case, the remainder is [tex]\(3\)[/tex].
So, the remainder when dividing [tex]\(x^3 + 2x^2 - 3x + 3\)[/tex] by [tex]\(x - 1\)[/tex] is [tex]\(3\)[/tex].
1. Identify the polynomial and divisor: We are dividing the polynomial [tex]\(x^3 + 2x^2 - 3x + 3\)[/tex] by [tex]\(x - 1\)[/tex]. In synthetic division, we use the root of the divisor, which in this case is [tex]\(1\)[/tex] (from [tex]\(x - 1 = 0\)[/tex]).
2. Set up the synthetic division process:
- Write down the coefficients of the polynomial: [tex]\(1, 2, -3, 3\)[/tex].
- Set aside the constant or root of the divisor, which is [tex]\(1\)[/tex].
3. Perform the synthetic division:
- Start with the leading coefficient, which is [tex]\(1\)[/tex]. Bring it straight down, as this is our initial result.
- Step 1: Multiply this result ([tex]\(1\)[/tex]) by the root of the divisor ([tex]\(1\)[/tex]) and write the result under the next coefficient. Add this result to the next coefficient:
[tex]\[
2 + (1 \times 1) = 2 + 1 = 3
\][/tex]
So, replace the second coefficient with [tex]\(3\)[/tex].
- Step 2: Multiply the result from Step 1 ([tex]\(3\)[/tex]) by the root of the divisor ([tex]\(1\)[/tex]):
[tex]\[
-3 + (3 \times 1) = -3 + 3 = 0
\][/tex]
So, replace the third coefficient with [tex]\(0\)[/tex].
- Step 3: Multiply the result from Step 2 ([tex]\(0\)[/tex]) by the root of the divisor ([tex]\(1\)[/tex]):
[tex]\[
3 + (0 \times 1) = 3 + 0 = 3
\][/tex]
So, replace the last coefficient with [tex]\(3\)[/tex].
4. Determine the remainder: The last number obtained after processing all coefficients is the remainder of the division. In this case, the remainder is [tex]\(3\)[/tex].
So, the remainder when dividing [tex]\(x^3 + 2x^2 - 3x + 3\)[/tex] by [tex]\(x - 1\)[/tex] is [tex]\(3\)[/tex].
