Answer :
To find the remainder when the polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by the polynomial [tex]\(x^2 + 3x + 3\)[/tex], we use polynomial long division.
### Step-by-Step Solution:
1. Divide the first term of the dividend [tex]\(3x^3\)[/tex] by the first term of the divisor [tex]\(x^2\)[/tex]:
[tex]\[
\frac{3x^3}{x^2} = 3x
\][/tex]
2. Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex]:
[tex]\[
3x \cdot (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x
\][/tex]
3. Subtract this product from the dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Now, divide [tex]\(-11x^2\)[/tex] by the first term of the divisor [tex]\(x^2\)[/tex]:
[tex]\[
\frac{-11x^2}{x^2} = -11
\][/tex]
5. Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(-11\)[/tex]:
[tex]\[
-11 \cdot (x^2 + 3x + 3) = -11x^2 - 33x - 33
\][/tex]
6. Subtract this product from the current polynomial:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
The remainder of the division is [tex]\(28x + 30\)[/tex].
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].
### Step-by-Step Solution:
1. Divide the first term of the dividend [tex]\(3x^3\)[/tex] by the first term of the divisor [tex]\(x^2\)[/tex]:
[tex]\[
\frac{3x^3}{x^2} = 3x
\][/tex]
2. Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex]:
[tex]\[
3x \cdot (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x
\][/tex]
3. Subtract this product from the dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Now, divide [tex]\(-11x^2\)[/tex] by the first term of the divisor [tex]\(x^2\)[/tex]:
[tex]\[
\frac{-11x^2}{x^2} = -11
\][/tex]
5. Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(-11\)[/tex]:
[tex]\[
-11 \cdot (x^2 + 3x + 3) = -11x^2 - 33x - 33
\][/tex]
6. Subtract this product from the current polynomial:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
The remainder of the division is [tex]\(28x + 30\)[/tex].
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].
