Answer :
To find the remainder when [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by [tex]\(x^2 + 3x + 3\)[/tex], we need to perform polynomial division. We'll follow the steps of polynomial long division:
1. Setup the Division:
The dividend is [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] and the divisor is [tex]\(x^2 + 3x + 3\)[/tex].
2. Perform the Division:
- First Term: Divide the first term of the dividend, [tex]\(3x^3\)[/tex], by the first term of the divisor, [tex]\(x^2\)[/tex]. This gives [tex]\(3x\)[/tex].
- Multiply and Subtract:
Multiply [tex]\(3x\)[/tex] by the entire divisor to get [tex]\(3x(x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x\)[/tex].
Subtract this from the dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
- Second Term: 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 [tex]\(-11\)[/tex].
- Multiply and Subtract:
Multiply [tex]\(-11\)[/tex] by the entire divisor to get [tex]\(-11(x^2 + 3x + 3) = -11x^2 - 33x - 33\)[/tex].
Subtract this from the new polynomial:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
3. Result:
The division results in a quotient of [tex]\(3x - 11\)[/tex] and a remainder of [tex]\(28x + 30\)[/tex]. Since the remainder should be of lower degree than the divisor, the remainder here is correct.
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 is [tex]\(\boxed{28x + 30}\)[/tex].
1. Setup the Division:
The dividend is [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] and the divisor is [tex]\(x^2 + 3x + 3\)[/tex].
2. Perform the Division:
- First Term: Divide the first term of the dividend, [tex]\(3x^3\)[/tex], by the first term of the divisor, [tex]\(x^2\)[/tex]. This gives [tex]\(3x\)[/tex].
- Multiply and Subtract:
Multiply [tex]\(3x\)[/tex] by the entire divisor to get [tex]\(3x(x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x\)[/tex].
Subtract this from the dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
- Second Term: 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 [tex]\(-11\)[/tex].
- Multiply and Subtract:
Multiply [tex]\(-11\)[/tex] by the entire divisor to get [tex]\(-11(x^2 + 3x + 3) = -11x^2 - 33x - 33\)[/tex].
Subtract this from the new polynomial:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
3. Result:
The division results in a quotient of [tex]\(3x - 11\)[/tex] and a remainder of [tex]\(28x + 30\)[/tex]. Since the remainder should be of lower degree than the divisor, the remainder here is correct.
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 is [tex]\(\boxed{28x + 30}\)[/tex].
