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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 perform polynomial division. Here's a detailed step-by-step guide:
1. Set up the division: We have a dividend [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] and a divisor [tex]\(x^2 + 3x + 3\)[/tex].
2. Divide the leading term: Take the leading term of the dividend, [tex]\(3x^3\)[/tex], and divide it by the leading term of the divisor, [tex]\(x^2\)[/tex]. This gives us [tex]\(3x\)[/tex].
3. Multiply and subtract: Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(3x(x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x\)[/tex]. Subtract this result from the original polynomial to get the new polynomial:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the process: Now take the new leading term [tex]\(-11x^2\)[/tex] and divide it by the leading term of the divisor [tex]\(x^2\)[/tex]. This gives us a quotient of [tex]\(-11\)[/tex].
5. Multiply and subtract again: Multiply the divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(-11\)[/tex] to get [tex]\(-11(x^2 + 3x + 3) = -11x^2 - 33x - 33\)[/tex]. Subtract this from [tex]\(-11x^2 - 5x - 3\)[/tex]:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
6. Result: The degree of the resulting polynomial [tex]\(28x + 30\)[/tex] is less than that of the divisor [tex]\(x^2 + 3x + 3\)[/tex], which means this is the remainder.
Therefore, 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: We have a dividend [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] and a divisor [tex]\(x^2 + 3x + 3\)[/tex].
2. Divide the leading term: Take the leading term of the dividend, [tex]\(3x^3\)[/tex], and divide it by the leading term of the divisor, [tex]\(x^2\)[/tex]. This gives us [tex]\(3x\)[/tex].
3. Multiply and subtract: Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(3x(x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x\)[/tex]. Subtract this result from the original polynomial to get the new polynomial:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the process: Now take the new leading term [tex]\(-11x^2\)[/tex] and divide it by the leading term of the divisor [tex]\(x^2\)[/tex]. This gives us a quotient of [tex]\(-11\)[/tex].
5. Multiply and subtract again: Multiply the divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(-11\)[/tex] to get [tex]\(-11(x^2 + 3x + 3) = -11x^2 - 33x - 33\)[/tex]. Subtract this from [tex]\(-11x^2 - 5x - 3\)[/tex]:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
6. Result: The degree of the resulting polynomial [tex]\(28x + 30\)[/tex] is less than that of the divisor [tex]\(x^2 + 3x + 3\)[/tex], which means this is the remainder.
Therefore, 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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