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], you can use polynomial division. Here’s how you can solve it:
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. Perform Polynomial Division:
- Divide the first term of the dividend [tex]\(3x^3\)[/tex] by the first term of the divisor [tex]\(x^2\)[/tex] to get the first term of the quotient, which is [tex]\(3x\)[/tex].
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by this term [tex]\(3x\)[/tex], which gives [tex]\(3x^3 + 9x^2 + 9x\)[/tex].
- Subtract this result from the dividend:
[tex]\( (3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) \)[/tex] gives [tex]\(-11x^2 - 5x - 3\)[/tex].
3. Continue the Division:
- Divide the first term of the new dividend [tex]\(-11x^2\)[/tex] by the first term of the divisor [tex]\(x^2\)[/tex] to get the next term of the quotient, which is [tex]\(-11\)[/tex].
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(-11\)[/tex], which gives [tex]\(-11x^2 - 33x - 33\)[/tex].
- Subtract this result from the current dividend:
[tex]\( (-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) \)[/tex] gives [tex]\(28x + 30\)[/tex].
4. Conclusion:
- 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:
- The dividend is [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex].
- The divisor is [tex]\(x^2 + 3x + 3\)[/tex].
2. Perform Polynomial Division:
- Divide the first term of the dividend [tex]\(3x^3\)[/tex] by the first term of the divisor [tex]\(x^2\)[/tex] to get the first term of the quotient, which is [tex]\(3x\)[/tex].
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by this term [tex]\(3x\)[/tex], which gives [tex]\(3x^3 + 9x^2 + 9x\)[/tex].
- Subtract this result from the dividend:
[tex]\( (3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) \)[/tex] gives [tex]\(-11x^2 - 5x - 3\)[/tex].
3. Continue the Division:
- Divide the first term of the new dividend [tex]\(-11x^2\)[/tex] by the first term of the divisor [tex]\(x^2\)[/tex] to get the next term of the quotient, which is [tex]\(-11\)[/tex].
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(-11\)[/tex], which gives [tex]\(-11x^2 - 33x - 33\)[/tex].
- Subtract this result from the current dividend:
[tex]\( (-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) \)[/tex] gives [tex]\(28x + 30\)[/tex].
4. Conclusion:
- 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].
