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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 division. Here's how the process works:
1. Setup the division: We have the dividend [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] and the divisor [tex]\(x^2 + 3x + 3\)[/tex].
2. Divide the leading terms: Divide the leading term of the dividend, [tex]\(3x^3\)[/tex], by the leading term of the divisor, [tex]\(x^2\)[/tex], which gives us [tex]\(3x\)[/tex].
3. Multiply and subtract: Multiply the entire divisor by [tex]\(3x\)[/tex] to get [tex]\(3x(x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x\)[/tex]. Subtract this from the original dividend to get:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the process: Now, divide the leading term [tex]\(-11x^2\)[/tex] by [tex]\(x^2\)[/tex], which gives us [tex]\(-11\)[/tex].
5. Multiply and subtract again: Multiply the divisor by [tex]\(-11\)[/tex] to get [tex]\(-11(x^2 + 3x + 3) = -11x^2 - 33x - 33\)[/tex]. Subtract this from the new dividend:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
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 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]\(\boxed{28x + 30}\)[/tex].
1. Setup the division: We have the dividend [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] and the divisor [tex]\(x^2 + 3x + 3\)[/tex].
2. Divide the leading terms: Divide the leading term of the dividend, [tex]\(3x^3\)[/tex], by the leading term of the divisor, [tex]\(x^2\)[/tex], which gives us [tex]\(3x\)[/tex].
3. Multiply and subtract: Multiply the entire divisor by [tex]\(3x\)[/tex] to get [tex]\(3x(x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x\)[/tex]. Subtract this from the original dividend to get:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the process: Now, divide the leading term [tex]\(-11x^2\)[/tex] by [tex]\(x^2\)[/tex], which gives us [tex]\(-11\)[/tex].
5. Multiply and subtract again: Multiply the divisor by [tex]\(-11\)[/tex] to get [tex]\(-11(x^2 + 3x + 3) = -11x^2 - 33x - 33\)[/tex]. Subtract this from the new dividend:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
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 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]\(\boxed{28x + 30}\)[/tex].
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