Answer :
To find the remainder when the polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by [tex]\(x^2 + 3x + 3\)[/tex], we will use polynomial division. Here’s a step-by-step walkthrough of the process:
1. Identify the Dividend and Divisor:
- Dividend: [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex]
- Divisor: [tex]\(x^2 + 3x + 3\)[/tex]
2. Set Up the Division:
Polynomial division is similar to long division with numbers. We'll divide the first term of the dividend by the first term of the divisor.
3. Divide the Leading Terms:
- Divide [tex]\(3x^3\)[/tex] by [tex]\(x^2\)[/tex]. This gives you [tex]\(3x\)[/tex].
4. Multiply and Subtract:
- Multiply [tex]\(3x\)[/tex] by the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] to get [tex]\(3x^3 + 9x^2 + 9x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
5. Repeat the Process:
- Divide the leading term [tex]\(-11x^2\)[/tex] by [tex]\(x^2\)[/tex] to get [tex]\(-11\)[/tex].
- Multiply [tex]\(-11\)[/tex] by the divisor, giving [tex]\(-11x^2 - 33x - 33\)[/tex].
- Subtract:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
6. Remainder:
The division process stops here because the remainder [tex]\(28x + 30\)[/tex] has a degree less than the divisor [tex]\(x^2 + 3x + 3\)[/tex].
Hence, the remainder when [tex]\((3x^3 - 2x^2 + 4x - 3)\)[/tex] is divided by [tex]\((x^2 + 3x + 3)\)[/tex] is [tex]\(28x + 30\)[/tex].
Therefore, the correct answer is [tex]\(\boxed{28x + 30}\)[/tex].
1. Identify the Dividend and Divisor:
- Dividend: [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex]
- Divisor: [tex]\(x^2 + 3x + 3\)[/tex]
2. Set Up the Division:
Polynomial division is similar to long division with numbers. We'll divide the first term of the dividend by the first term of the divisor.
3. Divide the Leading Terms:
- Divide [tex]\(3x^3\)[/tex] by [tex]\(x^2\)[/tex]. This gives you [tex]\(3x\)[/tex].
4. Multiply and Subtract:
- Multiply [tex]\(3x\)[/tex] by the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] to get [tex]\(3x^3 + 9x^2 + 9x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
5. Repeat the Process:
- Divide the leading term [tex]\(-11x^2\)[/tex] by [tex]\(x^2\)[/tex] to get [tex]\(-11\)[/tex].
- Multiply [tex]\(-11\)[/tex] by the divisor, giving [tex]\(-11x^2 - 33x - 33\)[/tex].
- Subtract:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
6. Remainder:
The division process stops here because the remainder [tex]\(28x + 30\)[/tex] has a degree less than the divisor [tex]\(x^2 + 3x + 3\)[/tex].
Hence, the remainder when [tex]\((3x^3 - 2x^2 + 4x - 3)\)[/tex] is divided by [tex]\((x^2 + 3x + 3)\)[/tex] is [tex]\(28x + 30\)[/tex].
Therefore, the correct answer is [tex]\(\boxed{28x + 30}\)[/tex].
