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Answer :
To find the remainder of the division [tex]\((3x^3 - 2x^2 + 4x - 3)\)[/tex] by [tex]\((x^2 + 3x + 3)\)[/tex], we perform polynomial division. Here is how you can think about the process:
1. Set Up the Division: We are dividing the polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
2. Perform the Division:
- The degree of the dividend is 3, and the degree of the divisor is 2. Since the divisor degree is less than or equal to the dividend degree, we'll divide normally.
- Start with the leading term of the dividend, [tex]\(3x^3\)[/tex]. Divide it by the leading term of the divisor, [tex]\(x^2\)[/tex], to get the first term of the quotient, which is [tex]\(3x\)[/tex].
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by this term [tex]\(3x\)[/tex] to get [tex]\(3x^3 + 9x^2 + 9x\)[/tex].
- Subtract this from the original polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] to find the new polynomial. Doing this, you get [tex]\(-11x^2 - 5x - 3\)[/tex].
- Now take the new leading term [tex]\(-11x^2\)[/tex] and divide it by [tex]\(x^2\)[/tex] to get the next term of the quotient, which is [tex]\(-11\)[/tex].
- Multiply the entire divisor by [tex]\(-11\)[/tex] to get [tex]\(-11x^2 - 33x - 33\)[/tex].
- Subtract this from the new dividend [tex]\(-11x^2 - 5x - 3\)[/tex], which leaves us with a new polynomial of [tex]\(28x + 30\)[/tex].
3. Resulting Remainder: After completing the division process, the remainder of the division is [tex]\(28x + 30\)[/tex].
So, 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]. Thus, the correct choice is [tex]\(28x + 30\)[/tex].
1. Set Up the Division: We are dividing the polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
2. Perform the Division:
- The degree of the dividend is 3, and the degree of the divisor is 2. Since the divisor degree is less than or equal to the dividend degree, we'll divide normally.
- Start with the leading term of the dividend, [tex]\(3x^3\)[/tex]. Divide it by the leading term of the divisor, [tex]\(x^2\)[/tex], to get the first term of the quotient, which is [tex]\(3x\)[/tex].
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by this term [tex]\(3x\)[/tex] to get [tex]\(3x^3 + 9x^2 + 9x\)[/tex].
- Subtract this from the original polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] to find the new polynomial. Doing this, you get [tex]\(-11x^2 - 5x - 3\)[/tex].
- Now take the new leading term [tex]\(-11x^2\)[/tex] and divide it by [tex]\(x^2\)[/tex] to get the next term of the quotient, which is [tex]\(-11\)[/tex].
- Multiply the entire divisor by [tex]\(-11\)[/tex] to get [tex]\(-11x^2 - 33x - 33\)[/tex].
- Subtract this from the new dividend [tex]\(-11x^2 - 5x - 3\)[/tex], which leaves us with a new polynomial of [tex]\(28x + 30\)[/tex].
3. Resulting Remainder: After completing the division process, the remainder of the division is [tex]\(28x + 30\)[/tex].
So, 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]. Thus, the correct choice is [tex]\(28x + 30\)[/tex].
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