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Answer :
When dividing the polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex], we follow the process of polynomial division. This is similar to long division with numbers. Here's how you can think about the steps involved:
1. Set up the division: We need to divide [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
2. Divide the leading terms: Start by dividing the leading term of the dividend, [tex]\(3x^3\)[/tex], 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], giving us [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]
4. Repeat the process: Now divide the new leading term [tex]\(-11x^2\)[/tex] by [tex]\(x^2\)[/tex], which yields [tex]\(-11\)[/tex].
5. Multiply and subtract again: Multiply the entire divisor by [tex]\(-11\)[/tex], giving [tex]\(-11x^2 - 33x - 33\)[/tex]. Subtract this from the current result:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
6. The remainder: When you cannot divide any further (because the degree of what's left, [tex]\(28x + 30\)[/tex], is less than the degree of the divisor, [tex]\(x^2 + 3x + 3\)[/tex]), you have your remainder.
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: We need to divide [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
2. Divide the leading terms: Start by dividing the leading term of the dividend, [tex]\(3x^3\)[/tex], 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], giving us [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]
4. Repeat the process: Now divide the new leading term [tex]\(-11x^2\)[/tex] by [tex]\(x^2\)[/tex], which yields [tex]\(-11\)[/tex].
5. Multiply and subtract again: Multiply the entire divisor by [tex]\(-11\)[/tex], giving [tex]\(-11x^2 - 33x - 33\)[/tex]. Subtract this from the current result:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
6. The remainder: When you cannot divide any further (because the degree of what's left, [tex]\(28x + 30\)[/tex], is less than the degree of the divisor, [tex]\(x^2 + 3x + 3\)[/tex]), you have your remainder.
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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