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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 can use polynomial long division. Here's a step-by-step explanation of the process:
1. Set Up the Division: Arrange the terms of both the dividend [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] and the divisor [tex]\(x^2 + 3x + 3\)[/tex] in decreasing order of powers.
2. Divide the First Term: Divide the first term of the dividend [tex]\(3x^3\)[/tex] by the first term of the divisor [tex]\(x^2\)[/tex] to get [tex]\(3x\)[/tex]. This will be the first term of the quotient.
3. Multiply and Subtract: Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex] (the term we calculated in step 2). This gives:
[tex]\[
3x \times (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x
\][/tex]
Subtract this result from the original dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the Process: Now, treat [tex]\(-11x^2 - 5x - 3\)[/tex] as the new dividend. Divide the first term [tex]\(-11x^2\)[/tex] by the first term of the divisor [tex]\(x^2\)[/tex] to get [tex]\(-11\)[/tex].
5. Multiply and Subtract Again: Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(-11\)[/tex]:
[tex]\[
-11 \times (x^2 + 3x + 3) = -11x^2 - 33x - 33
\][/tex]
Now, subtract this from the current dividend:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
6. Conclusion: The process stops here because the degree of the remainder [tex]\(28x + 30\)[/tex] is less than the degree of the divisor [tex]\(x^2 + 3x + 3\)[/tex]. Therefore, the remainder of the division is [tex]\(28x + 30\)[/tex].
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].
1. Set Up the Division: Arrange the terms of both the dividend [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] and the divisor [tex]\(x^2 + 3x + 3\)[/tex] in decreasing order of powers.
2. Divide the First Term: Divide the first term of the dividend [tex]\(3x^3\)[/tex] by the first term of the divisor [tex]\(x^2\)[/tex] to get [tex]\(3x\)[/tex]. This will be the first term of the quotient.
3. Multiply and Subtract: Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex] (the term we calculated in step 2). This gives:
[tex]\[
3x \times (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x
\][/tex]
Subtract this result from the original dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the Process: Now, treat [tex]\(-11x^2 - 5x - 3\)[/tex] as the new dividend. Divide the first term [tex]\(-11x^2\)[/tex] by the first term of the divisor [tex]\(x^2\)[/tex] to get [tex]\(-11\)[/tex].
5. Multiply and Subtract Again: Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(-11\)[/tex]:
[tex]\[
-11 \times (x^2 + 3x + 3) = -11x^2 - 33x - 33
\][/tex]
Now, subtract this from the current dividend:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
6. Conclusion: The process stops here because the degree of the remainder [tex]\(28x + 30\)[/tex] is less than the degree of the divisor [tex]\(x^2 + 3x + 3\)[/tex]. Therefore, the remainder of the division is [tex]\(28x + 30\)[/tex].
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].
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