Answer :
To find the remainder when dividing [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex], we need to use polynomial long division. Let's work through the steps:
1. Setup the Division: We will divide [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
2. Divide the Leading Terms: Divide the leading term of the dividend, [tex]\(3x^3\)[/tex], by the leading term of the divisor, [tex]\(x^2\)[/tex]. This gives [tex]\(3x\)[/tex].
3. Multiply and Subtract: Multiply [tex]\(3x\)[/tex] by the entire divisor [tex]\((x^2 + 3x + 3)\)[/tex], which gives [tex]\(3x^3 + 9x^2 + 9x\)[/tex]. Now subtract this from the original polynomial:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the Process: Now divide the new leading term [tex]\(-11x^2\)[/tex] by the leading term of the divisor, [tex]\(x^2\)[/tex], which gives [tex]\(-11\)[/tex].
5. Multiply and Subtract Again: Multiply [tex]\(-11\)[/tex] by the entire divisor, resulting in [tex]\(-11x^2 - 33x - 33\)[/tex]. Subtract this from [tex]\(-11x^2 - 5x - 3\)[/tex]:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
6. Identify the Remainder: The result of [tex]\(28x + 30\)[/tex] is the remainder, because it has a lower degree than the divisor [tex]\(x^2 + 3x + 3\)[/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].
Thus, the correct answer is [tex]\(\boxed{28x + 30}\)[/tex].
1. Setup the Division: We will divide [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
2. Divide the Leading Terms: Divide the leading term of the dividend, [tex]\(3x^3\)[/tex], by the leading term of the divisor, [tex]\(x^2\)[/tex]. This gives [tex]\(3x\)[/tex].
3. Multiply and Subtract: Multiply [tex]\(3x\)[/tex] by the entire divisor [tex]\((x^2 + 3x + 3)\)[/tex], which gives [tex]\(3x^3 + 9x^2 + 9x\)[/tex]. Now subtract this from the original polynomial:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the Process: Now divide the new leading term [tex]\(-11x^2\)[/tex] by the leading term of the divisor, [tex]\(x^2\)[/tex], which gives [tex]\(-11\)[/tex].
5. Multiply and Subtract Again: Multiply [tex]\(-11\)[/tex] by the entire divisor, resulting in [tex]\(-11x^2 - 33x - 33\)[/tex]. Subtract this from [tex]\(-11x^2 - 5x - 3\)[/tex]:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
6. Identify the Remainder: The result of [tex]\(28x + 30\)[/tex] is the remainder, because it has a lower degree than the divisor [tex]\(x^2 + 3x + 3\)[/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].
Thus, the correct answer is [tex]\(\boxed{28x + 30}\)[/tex].
