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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 perform polynomial division. Here's a detailed, step-by-step solution:
1. Setup the Division:
- Divide the polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
- Write it in the form of a long division where [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is under the division symbol and [tex]\(x^2 + 3x + 3\)[/tex] is outside.
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], which gives [tex]\(3x\)[/tex].
- Multiply [tex]\(3x\)[/tex] by the whole divisor [tex]\(x^2 + 3x + 3\)[/tex].
3. Subtract and Bring Down:
- Subtract the result of the multiplication from the original polynomial.
- This subtraction gives a new polynomial: the remainder of the division so far. Bring down the next term if there is one.
4. Repeat the Process:
- Divide the new leading term by the leading term of the divisor.
- This time, divide the new leading term by [tex]\(x^2\)[/tex] again. Continue this process until the degree of the new remainder is less than the degree of the divisor.
5. Identify the Remainder:
- Once you cannot divide anymore (because the degree of what's left is less than the degree of [tex]\(x^2 + 3x + 3\)[/tex]), what's left is the remainder.
Through this method, you would find that the remainder of the division 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].
1. Setup the Division:
- Divide the polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
- Write it in the form of a long division where [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is under the division symbol and [tex]\(x^2 + 3x + 3\)[/tex] is outside.
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], which gives [tex]\(3x\)[/tex].
- Multiply [tex]\(3x\)[/tex] by the whole divisor [tex]\(x^2 + 3x + 3\)[/tex].
3. Subtract and Bring Down:
- Subtract the result of the multiplication from the original polynomial.
- This subtraction gives a new polynomial: the remainder of the division so far. Bring down the next term if there is one.
4. Repeat the Process:
- Divide the new leading term by the leading term of the divisor.
- This time, divide the new leading term by [tex]\(x^2\)[/tex] again. Continue this process until the degree of the new remainder is less than the degree of the divisor.
5. Identify the Remainder:
- Once you cannot divide anymore (because the degree of what's left is less than the degree of [tex]\(x^2 + 3x + 3\)[/tex]), what's left is the remainder.
Through this method, you would find that the remainder of the division 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].
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