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Answer :
Sure! 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's a step-by-step guide on how to perform this division:
1. Setup the Division: Divide the polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by the polynomial [tex]\(x^2 + 3x + 3\)[/tex].
2. First Division Step:
- Divide the first term of the dividend [tex]\(3x^3\)[/tex] by the first term of the divisor [tex]\(x^2\)[/tex], which gives [tex]\(3x\)[/tex].
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/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]
3. Second Division Step:
- Divide the new first term [tex]\(-11x^2\)[/tex] by [tex]\(x^2\)[/tex], which gives [tex]\(-11\)[/tex].
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(-11\)[/tex] to get [tex]\(-11x^2 - 33x - 33\)[/tex].
- Subtract from the previous result:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
4. Conclusion:
- The remainder, after division, is [tex]\(28x + 30\)[/tex].
Therefore, the remainder when [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by [tex]\(x^2 + 3x + 3\)[/tex] is [tex]\(\boxed{28x + 30}\)[/tex].
1. Setup the Division: Divide the polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by the polynomial [tex]\(x^2 + 3x + 3\)[/tex].
2. First Division Step:
- Divide the first term of the dividend [tex]\(3x^3\)[/tex] by the first term of the divisor [tex]\(x^2\)[/tex], which gives [tex]\(3x\)[/tex].
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/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]
3. Second Division Step:
- Divide the new first term [tex]\(-11x^2\)[/tex] by [tex]\(x^2\)[/tex], which gives [tex]\(-11\)[/tex].
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(-11\)[/tex] to get [tex]\(-11x^2 - 33x - 33\)[/tex].
- Subtract from the previous result:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
4. Conclusion:
- The remainder, after division, is [tex]\(28x + 30\)[/tex].
Therefore, the remainder when [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by [tex]\(x^2 + 3x + 3\)[/tex] is [tex]\(\boxed{28x + 30}\)[/tex].
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