Answer :
To find the remainder when dividing the polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex], we use polynomial long division or synthetic division. Here's a step-by-step solution using polynomial long division:
1. Setup the Division: Write down 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]. This gives us [tex]\(3x\)[/tex].
3. Multiply and Subtract:
- Multiply [tex]\(3x\)[/tex] by the entire divisor [tex]\((x^2 + 3x + 3)\)[/tex], resulting in [tex]\(3x^3 + 9x^2 + 9x\)[/tex].
- Subtract this from the current dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the Process:
- Take the leading term of the new polynomial [tex]\(-11x^2\)[/tex] and divide by the leading term of the divisor [tex]\(x^2\)[/tex], resulting in [tex]\(-11\)[/tex].
- Multiply [tex]\(-11\)[/tex] by the entire divisor [tex]\((x^2 + 3x + 3)\)[/tex], resulting in [tex]\(-11x^2 - 33x - 33\)[/tex].
- Subtract this from the polynomial:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
5. Conclusion:
- Since the degree of the resulting polynomial [tex]\(28x + 30\)[/tex] is less than the degree of the divisor [tex]\(x^2 + 3x + 3\)[/tex], this is the remainder of the division.
Therefore, 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: Write down 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]. This gives us [tex]\(3x\)[/tex].
3. Multiply and Subtract:
- Multiply [tex]\(3x\)[/tex] by the entire divisor [tex]\((x^2 + 3x + 3)\)[/tex], resulting in [tex]\(3x^3 + 9x^2 + 9x\)[/tex].
- Subtract this from the current dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the Process:
- Take the leading term of the new polynomial [tex]\(-11x^2\)[/tex] and divide by the leading term of the divisor [tex]\(x^2\)[/tex], resulting in [tex]\(-11\)[/tex].
- Multiply [tex]\(-11\)[/tex] by the entire divisor [tex]\((x^2 + 3x + 3)\)[/tex], resulting in [tex]\(-11x^2 - 33x - 33\)[/tex].
- Subtract this from the polynomial:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
5. Conclusion:
- Since the degree of the resulting polynomial [tex]\(28x + 30\)[/tex] is less than the degree of the divisor [tex]\(x^2 + 3x + 3\)[/tex], this is the remainder of the division.
Therefore, 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].
