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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 need to perform polynomial division. Here's how it's done step-by-step:
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:
We are dividing a cubic polynomial by a quadratic polynomial, so the remainder will be a linear polynomial at most, of the form [tex]\(ax + b\)[/tex].
3. Perform Polynomial Division:
- Divide the leading term of the dividend [tex]\(3x^3\)[/tex] by the leading term of the divisor [tex]\(x^2\)[/tex] to get [tex]\(3x\)[/tex].
- Multiply the whole divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex] and subtract the result from the original dividend.
- Repeat the process with the new polynomial obtained after subtraction. The first term will now be a quadratic polynomial.
- Continue the process until the degree of the remainder polynomial is less than the degree of the divisor polynomial.
4. Determine the Remainder:
After performing these steps, any terms left that do not divide further by the divisor (because they have lower degree than the divisor) will form the remainder.
The remainder, after completing these steps, 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]\(28x + 30\)[/tex].
So, the correct answer to the question is [tex]\(28x + 30\)[/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:
We are dividing a cubic polynomial by a quadratic polynomial, so the remainder will be a linear polynomial at most, of the form [tex]\(ax + b\)[/tex].
3. Perform Polynomial Division:
- Divide the leading term of the dividend [tex]\(3x^3\)[/tex] by the leading term of the divisor [tex]\(x^2\)[/tex] to get [tex]\(3x\)[/tex].
- Multiply the whole divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex] and subtract the result from the original dividend.
- Repeat the process with the new polynomial obtained after subtraction. The first term will now be a quadratic polynomial.
- Continue the process until the degree of the remainder polynomial is less than the degree of the divisor polynomial.
4. Determine the Remainder:
After performing these steps, any terms left that do not divide further by the divisor (because they have lower degree than the divisor) will form the remainder.
The remainder, after completing these steps, 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]\(28x + 30\)[/tex].
So, the correct answer to the question is [tex]\(28x + 30\)[/tex].
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