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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:
1. Set up the division: We are dividing [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
2. First term division:
- Divide the leading term of the dividend ([tex]\(3x^3\)[/tex]) by the leading term of the divisor ([tex]\(x^2\)[/tex]):
[tex]\[
\frac{3x^3}{x^2} = 3x
\][/tex]
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex] and subtract the result from the dividend:
[tex]\[
\text{Multiply: } (x^2 + 3x + 3) \cdot 3x = 3x^3 + 9x^2 + 9x
\][/tex]
Subtract from the original polynomial:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
3. Next term division:
- Divide the new leading term ([tex]\(-11x^2\)[/tex]) by the leading term of the divisor ([tex]\(x^2\)[/tex]):
[tex]\[
\frac{-11x^2}{x^2} = -11
\][/tex]
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(-11\)[/tex] and subtract:
[tex]\[
\text{Multiply: } (x^2 + 3x + 3) \cdot (-11) = -11x^2 - 33x - 33
\][/tex]
Subtract this from the current polynomial:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
4. Remainder: Since the degree of the new polynomial [tex]\(28x + 30\)[/tex] is less than the degree of the divisor ([tex]\(x^2 + 3x + 3\)[/tex]), we stop here. The remainder is [tex]\(28x + 30\)[/tex].
Thus, when [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by [tex]\(x^2 + 3x + 3\)[/tex], the remainder is [tex]\(\boxed{28x + 30}\)[/tex].
1. Set up the division: We are dividing [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
2. First term division:
- Divide the leading term of the dividend ([tex]\(3x^3\)[/tex]) by the leading term of the divisor ([tex]\(x^2\)[/tex]):
[tex]\[
\frac{3x^3}{x^2} = 3x
\][/tex]
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex] and subtract the result from the dividend:
[tex]\[
\text{Multiply: } (x^2 + 3x + 3) \cdot 3x = 3x^3 + 9x^2 + 9x
\][/tex]
Subtract from the original polynomial:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
3. Next term division:
- Divide the new leading term ([tex]\(-11x^2\)[/tex]) by the leading term of the divisor ([tex]\(x^2\)[/tex]):
[tex]\[
\frac{-11x^2}{x^2} = -11
\][/tex]
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(-11\)[/tex] and subtract:
[tex]\[
\text{Multiply: } (x^2 + 3x + 3) \cdot (-11) = -11x^2 - 33x - 33
\][/tex]
Subtract this from the current polynomial:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
4. Remainder: Since the degree of the new polynomial [tex]\(28x + 30\)[/tex] is less than the degree of the divisor ([tex]\(x^2 + 3x + 3\)[/tex]), we stop here. The remainder is [tex]\(28x + 30\)[/tex].
Thus, when [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by [tex]\(x^2 + 3x + 3\)[/tex], the remainder is [tex]\(\boxed{28x + 30}\)[/tex].
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