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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], we need to perform polynomial division. This process is similar to long division with numbers, but instead, we use terms involving [tex]\(x\)[/tex].
### Step-by-Step Solution:
1. Set Up the Division:
- We have the dividend: [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex].
- The divisor is [tex]\(x^2 + 3x + 3\)[/tex].
2. Division Process:
- First Division:
- Divide the leading term of the dividend [tex]\(3x^3\)[/tex] by the leading term of the divisor [tex]\(x^2\)[/tex], which gives [tex]\(3x\)[/tex].
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex], which results in [tex]\(3x^3 + 9x^2 + 9x\)[/tex].
- Subtract this result from the original polynomial:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
- Second Division:
- Divide the new leading term [tex]\(-11x^2\)[/tex] by [tex]\(x^2\)[/tex], which gives [tex]\(-11\)[/tex].
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(-11\)[/tex], which results in [tex]\(-11x^2 - 33x - 33\)[/tex].
- Subtract this result from [tex]\(-11x^2 - 5x - 3\)[/tex]:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
3. Determine the Remainder:
- After the second division, we are left with [tex]\(28x + 30\)[/tex], which is of lower degree than the divisor. This expression is the remainder of the division.
Thus, 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]. Therefore, the answer is [tex]\(28x + 30\)[/tex].
### Step-by-Step Solution:
1. Set Up the Division:
- We have the dividend: [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex].
- The divisor is [tex]\(x^2 + 3x + 3\)[/tex].
2. Division Process:
- First Division:
- Divide the leading term of the dividend [tex]\(3x^3\)[/tex] by the leading term of the divisor [tex]\(x^2\)[/tex], which gives [tex]\(3x\)[/tex].
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex], which results in [tex]\(3x^3 + 9x^2 + 9x\)[/tex].
- Subtract this result from the original polynomial:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
- Second Division:
- Divide the new leading term [tex]\(-11x^2\)[/tex] by [tex]\(x^2\)[/tex], which gives [tex]\(-11\)[/tex].
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(-11\)[/tex], which results in [tex]\(-11x^2 - 33x - 33\)[/tex].
- Subtract this result from [tex]\(-11x^2 - 5x - 3\)[/tex]:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
3. Determine the Remainder:
- After the second division, we are left with [tex]\(28x + 30\)[/tex], which is of lower degree than the divisor. This expression is the remainder of the division.
Thus, 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]. Therefore, the answer is [tex]\(28x + 30\)[/tex].
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