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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 use polynomial long division.
### Step-by-Step Solution:
1. Divide the first term of the dividend by the first term of the divisor:
- The first term of the dividend is [tex]\(3x^3\)[/tex] and the first term of the divisor is [tex]\(x^2\)[/tex].
- Divide [tex]\(3x^3\)[/tex] by [tex]\(x^2\)[/tex] to get [tex]\(3x\)[/tex].
2. Multiply the entire divisor by this term:
- Multiply [tex]\(3x\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
- This gives you [tex]\(3x \cdot (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x\)[/tex].
3. Subtract this from the original polynomial:
- Subtract [tex]\(3x^3 + 9x^2 + 9x\)[/tex] from [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex].
- The result is: [tex]\((-2x^2 + 4x - 3) - (9x^2 + 9x) = -11x^2 - 5x - 3\)[/tex].
4. Repeat the process with the new polynomial:
- Divide the first term of the new polynomial, [tex]\(-11x^2\)[/tex], by the first term of the divisor, [tex]\(x^2\)[/tex].
- This gives you [tex]\(-11\)[/tex].
5. Multiply the entire divisor by [tex]\(-11\)[/tex]:
- Multiply [tex]\(-11\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
- This gives you [tex]\(-11 \cdot (x^2 + 3x + 3) = -11x^2 - 33x - 33\)[/tex].
6. Subtract this from the current polynomial:
- Subtract [tex]\(-11x^2 - 33x - 33\)[/tex] from [tex]\(-11x^2 - 5x - 3\)[/tex].
- The result is: [tex]\((-11x^2 - 5x - 3) - (-11x^2 - 33x - 33)\)[/tex].
- Simplify to get [tex]\(28x + 30\)[/tex].
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 answer is [tex]\(28x + 30\)[/tex].
### Step-by-Step Solution:
1. Divide the first term of the dividend by the first term of the divisor:
- The first term of the dividend is [tex]\(3x^3\)[/tex] and the first term of the divisor is [tex]\(x^2\)[/tex].
- Divide [tex]\(3x^3\)[/tex] by [tex]\(x^2\)[/tex] to get [tex]\(3x\)[/tex].
2. Multiply the entire divisor by this term:
- Multiply [tex]\(3x\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
- This gives you [tex]\(3x \cdot (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x\)[/tex].
3. Subtract this from the original polynomial:
- Subtract [tex]\(3x^3 + 9x^2 + 9x\)[/tex] from [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex].
- The result is: [tex]\((-2x^2 + 4x - 3) - (9x^2 + 9x) = -11x^2 - 5x - 3\)[/tex].
4. Repeat the process with the new polynomial:
- Divide the first term of the new polynomial, [tex]\(-11x^2\)[/tex], by the first term of the divisor, [tex]\(x^2\)[/tex].
- This gives you [tex]\(-11\)[/tex].
5. Multiply the entire divisor by [tex]\(-11\)[/tex]:
- Multiply [tex]\(-11\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
- This gives you [tex]\(-11 \cdot (x^2 + 3x + 3) = -11x^2 - 33x - 33\)[/tex].
6. Subtract this from the current polynomial:
- Subtract [tex]\(-11x^2 - 33x - 33\)[/tex] from [tex]\(-11x^2 - 5x - 3\)[/tex].
- The result is: [tex]\((-11x^2 - 5x - 3) - (-11x^2 - 33x - 33)\)[/tex].
- Simplify to get [tex]\(28x + 30\)[/tex].
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 answer is [tex]\(28x + 30\)[/tex].
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