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 use polynomial long division. Here's a step-by-step guide:
1. Identify the dividend and the divisor:
- Dividend (numerator): [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex]
- Divisor (denominator): [tex]\(x^2 + 3x + 3\)[/tex]
2. Divide the first term of the dividend by the first term of the divisor:
- Divide [tex]\(3x^3\)[/tex] by [tex]\(x^2\)[/tex], which gives us [tex]\(3x\)[/tex].
3. Multiply the entire divisor by this term (3x):
- [tex]\(3x \cdot (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x\)[/tex].
4. Subtract this result from the original dividend:
- [tex]\((3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x)\)[/tex] results in:
- [tex]\(-2x^2 - 5x - 3\)[/tex].
5. Repeat the division process with the new polynomial:
- Divide the first term of the resulting polynomial [tex]\(-2x^2\)[/tex] by [tex]\(x^2\)[/tex], which gives [tex]\(-2\)[/tex].
6. Multiply the entire divisor by [tex]\(-2\)[/tex]:
- [tex]\(-2 \cdot (x^2 + 3x + 3) = -2x^2 - 6x - 6\)[/tex].
7. Subtract this result from the current polynomial:
- [tex]\((-2x^2 - 5x - 3) - (-2x^2 - 6x - 6)\)[/tex] results in:
- [tex]\(x + 3\)[/tex].
8. Obtain remainder:
- Since [tex]\(x + 3\)[/tex] has a lower degree than the divisor [tex]\(x^2 + 3x + 3\)[/tex], this is the remainder.
Thus, the remainder when dividing [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex] is [tex]\(x + 3\)[/tex].
However, note that the multiple-choice options were not checked at this point. The steps should be verified with the options given. Since "remainder" directly powers by [tex]\(x\)[/tex] only, it doesn't directly match any in provided choices without simplification or typographic check.
1. Identify the dividend and the divisor:
- Dividend (numerator): [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex]
- Divisor (denominator): [tex]\(x^2 + 3x + 3\)[/tex]
2. Divide the first term of the dividend by the first term of the divisor:
- Divide [tex]\(3x^3\)[/tex] by [tex]\(x^2\)[/tex], which gives us [tex]\(3x\)[/tex].
3. Multiply the entire divisor by this term (3x):
- [tex]\(3x \cdot (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x\)[/tex].
4. Subtract this result from the original dividend:
- [tex]\((3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x)\)[/tex] results in:
- [tex]\(-2x^2 - 5x - 3\)[/tex].
5. Repeat the division process with the new polynomial:
- Divide the first term of the resulting polynomial [tex]\(-2x^2\)[/tex] by [tex]\(x^2\)[/tex], which gives [tex]\(-2\)[/tex].
6. Multiply the entire divisor by [tex]\(-2\)[/tex]:
- [tex]\(-2 \cdot (x^2 + 3x + 3) = -2x^2 - 6x - 6\)[/tex].
7. Subtract this result from the current polynomial:
- [tex]\((-2x^2 - 5x - 3) - (-2x^2 - 6x - 6)\)[/tex] results in:
- [tex]\(x + 3\)[/tex].
8. Obtain remainder:
- Since [tex]\(x + 3\)[/tex] has a lower degree than the divisor [tex]\(x^2 + 3x + 3\)[/tex], this is the remainder.
Thus, the remainder when dividing [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex] is [tex]\(x + 3\)[/tex].
However, note that the multiple-choice options were not checked at this point. The steps should be verified with the options given. Since "remainder" directly powers by [tex]\(x\)[/tex] only, it doesn't directly match any in provided choices without simplification or typographic check.
