Answer :
To find the remainder when [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by [tex]\(x^2 + 3x + 3\)[/tex], we can use polynomial long division. Here is a step-by-step solution:
1. Set Up the Division:
- Dividend: [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex]
- Divisor: [tex]\(x^2 + 3x + 3\)[/tex]
2. Divide the First Terms:
- Divide the leading term of the dividend [tex]\(3x^3\)[/tex] by the leading term of the divisor [tex]\(x^2\)[/tex].
- This gives: [tex]\(3x\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor by [tex]\(3x\)[/tex]:
[tex]\[
3x \times (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x
\][/tex]
- Subtract this result from the original polynomial:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the Division:
- Divide the new leading term [tex]\(-11x^2\)[/tex] by the leading term of the divisor [tex]\(x^2\)[/tex]:
[tex]\(-11\)[/tex].
- Multiply and subtract:
[tex]\[
-11 \times (x^2 + 3x + 3) = -11x^2 - 33x - 33
\][/tex]
- Subtract this from [tex]\(-11x^2 - 5x - 3\)[/tex]:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
5. Conclusion:
- The remainder is [tex]\(28x + 30\)[/tex].
So, 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]\(28x + 30\)[/tex].
1. Set Up the Division:
- Dividend: [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex]
- Divisor: [tex]\(x^2 + 3x + 3\)[/tex]
2. Divide the First Terms:
- Divide the leading term of the dividend [tex]\(3x^3\)[/tex] by the leading term of the divisor [tex]\(x^2\)[/tex].
- This gives: [tex]\(3x\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor by [tex]\(3x\)[/tex]:
[tex]\[
3x \times (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x
\][/tex]
- Subtract this result from the original polynomial:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the Division:
- Divide the new leading term [tex]\(-11x^2\)[/tex] by the leading term of the divisor [tex]\(x^2\)[/tex]:
[tex]\(-11\)[/tex].
- Multiply and subtract:
[tex]\[
-11 \times (x^2 + 3x + 3) = -11x^2 - 33x - 33
\][/tex]
- Subtract this from [tex]\(-11x^2 - 5x - 3\)[/tex]:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
5. Conclusion:
- The remainder is [tex]\(28x + 30\)[/tex].
So, 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]\(28x + 30\)[/tex].
