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 long division. Here's a step-by-step explanation:
1. Set up the division: We want to divide the polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
2. Divide the first 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 [tex]\(3x\)[/tex].
3. Multiply and subtract:
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(3x^3 + 9x^2 + 9x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the division:
- Divide the new leading term [tex]\(-11x^2\)[/tex] by the leading term of the divisor [tex]\(x^2\)[/tex], which gives [tex]\(-11\)[/tex].
5. Multiply and subtract again:
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(-11\)[/tex] to obtain [tex]\(-11x^2 - 33x - 33\)[/tex].
- Subtract this from the current remainder:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
6. Result:
- Since the degree of the remainder [tex]\(28x + 30\)[/tex] is less than the degree of the divisor [tex]\(x^2 + 3x + 3\)[/tex], the process stops here.
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 want to divide the polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] by [tex]\(x^2 + 3x + 3\)[/tex].
2. Divide the first 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 [tex]\(3x\)[/tex].
3. Multiply and subtract:
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(3x^3 + 9x^2 + 9x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the division:
- Divide the new leading term [tex]\(-11x^2\)[/tex] by the leading term of the divisor [tex]\(x^2\)[/tex], which gives [tex]\(-11\)[/tex].
5. Multiply and subtract again:
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(-11\)[/tex] to obtain [tex]\(-11x^2 - 33x - 33\)[/tex].
- Subtract this from the current remainder:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
6. Result:
- Since the degree of the remainder [tex]\(28x + 30\)[/tex] is less than the degree of the divisor [tex]\(x^2 + 3x + 3\)[/tex], the process stops here.
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].
