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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 long division.
### Step-by-Step Solution
#### Step 1: Set Up Division
- Dividend: [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex]
- Divisor: [tex]\(x^2 + 3x + 3\)[/tex]
The degree of the dividend is 3, and the degree of the divisor is 2, so we will perform the division until we reduce the expression of a degree less than 2 (the degree of the divisor).
#### Step 2: Perform Polynomial Long Division
1. Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\frac{3x^3}{x^2} = 3x
\][/tex]
2. Multiply the entire divisor by this quotient term (3x):
[tex]\[
3x \times (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x
\][/tex]
3. Subtract this from the dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the process with the new polynomial [tex]\(-11x^2 - 5x - 3\)[/tex]:
[tex]\[
\frac{-11x^2}{x^2} = -11
\][/tex]
5. Multiply the entire divisor by [tex]\(-11\)[/tex]:
[tex]\[
-11 \times (x^2 + 3x + 3) = -11x^2 - 33x - 33
\][/tex]
6. Subtract this from the current polynomial:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
#### Final Result
The final result of the polynomial division is:
- Quotient: [tex]\(3x - 11\)[/tex]
- Remainder: [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]\(\boxed{28x + 30}\)[/tex].
### Step-by-Step Solution
#### Step 1: Set Up Division
- Dividend: [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex]
- Divisor: [tex]\(x^2 + 3x + 3\)[/tex]
The degree of the dividend is 3, and the degree of the divisor is 2, so we will perform the division until we reduce the expression of a degree less than 2 (the degree of the divisor).
#### Step 2: Perform Polynomial Long Division
1. Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\frac{3x^3}{x^2} = 3x
\][/tex]
2. Multiply the entire divisor by this quotient term (3x):
[tex]\[
3x \times (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x
\][/tex]
3. Subtract this from the dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the process with the new polynomial [tex]\(-11x^2 - 5x - 3\)[/tex]:
[tex]\[
\frac{-11x^2}{x^2} = -11
\][/tex]
5. Multiply the entire divisor by [tex]\(-11\)[/tex]:
[tex]\[
-11 \times (x^2 + 3x + 3) = -11x^2 - 33x - 33
\][/tex]
6. Subtract this from the current polynomial:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
#### Final Result
The final result of the polynomial division is:
- Quotient: [tex]\(3x - 11\)[/tex]
- Remainder: [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]\(\boxed{28x + 30}\)[/tex].
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