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Answer :
To find the remainder when dividing the polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex], we can use polynomial long division. Here is a detailed step-by-step approach:
1. Setup the Division:
- The dividend is [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex].
- The divisor is [tex]\(x^2 + 3x + 3\)[/tex].
2. Divide the Leading Terms:
- Divide the leading term of the dividend [tex]\(3x^3\)[/tex] by the leading term of the divisor [tex]\(x^2\)[/tex], which gives [tex]\(3x\)[/tex].
- Multiply the entire divisor by [tex]\(3x\)[/tex], resulting in [tex]\(3x(x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x\)[/tex].
3. Subtract and Bring Down:
- Subtract [tex]\(3x^3 + 9x^2 + 9x\)[/tex] from the dividend.
- The result is:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
- Bring down the next term if there is one. In this case, we already have the complete new polynomial [tex]\(-11x^2 - 5x - 3\)[/tex].
4. Repeat the Process:
- Now, divide the new leading term [tex]\(-11x^2\)[/tex] by the leading term of the divisor [tex]\(x^2\)[/tex], which gives [tex]\(-11\)[/tex].
- Multiply the entire divisor by [tex]\(-11\)[/tex], resulting in [tex]\(-11(x^2 + 3x + 3) = -11x^2 - 33x - 33\)[/tex].
5. Subtract Again:
- Subtract [tex]\(-11x^2 - 33x - 33\)[/tex] from [tex]\(-11x^2 - 5x - 3\)[/tex].
- The result is:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
6. Conclusion:
- Since the degree of the remainder [tex]\(28x + 30\)[/tex] is less than the degree of the divisor, the division process stops here.
- Therefore, the remainder when dividing [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex] is [tex]\(28x + 30\)[/tex].
So, the remainder is [tex]\(\boxed{28x + 30}\)[/tex].
1. Setup the Division:
- The dividend is [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex].
- The divisor is [tex]\(x^2 + 3x + 3\)[/tex].
2. Divide the Leading Terms:
- Divide the leading term of the dividend [tex]\(3x^3\)[/tex] by the leading term of the divisor [tex]\(x^2\)[/tex], which gives [tex]\(3x\)[/tex].
- Multiply the entire divisor by [tex]\(3x\)[/tex], resulting in [tex]\(3x(x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x\)[/tex].
3. Subtract and Bring Down:
- Subtract [tex]\(3x^3 + 9x^2 + 9x\)[/tex] from the dividend.
- The result is:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
- Bring down the next term if there is one. In this case, we already have the complete new polynomial [tex]\(-11x^2 - 5x - 3\)[/tex].
4. Repeat the Process:
- Now, divide the new leading term [tex]\(-11x^2\)[/tex] by the leading term of the divisor [tex]\(x^2\)[/tex], which gives [tex]\(-11\)[/tex].
- Multiply the entire divisor by [tex]\(-11\)[/tex], resulting in [tex]\(-11(x^2 + 3x + 3) = -11x^2 - 33x - 33\)[/tex].
5. Subtract Again:
- Subtract [tex]\(-11x^2 - 33x - 33\)[/tex] from [tex]\(-11x^2 - 5x - 3\)[/tex].
- The result is:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
6. Conclusion:
- Since the degree of the remainder [tex]\(28x + 30\)[/tex] is less than the degree of the divisor, the division process stops here.
- Therefore, the remainder when dividing [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex] is [tex]\(28x + 30\)[/tex].
So, the remainder is [tex]\(\boxed{28x + 30}\)[/tex].
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