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'll perform polynomial division. Here’s a step-by-step breakdown:
1. Setup the Division:
- We have the dividend as [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex].
- The divisor is [tex]\(x^2 + 3x + 3\)[/tex].
2. Division Process:
- First term of the quotient: 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].
- Multiply [tex]\(3x\)[/tex] by the entire divisor [tex]\((x^2 + 3x + 3)\)[/tex]:
[tex]\[
3x \times (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x
\][/tex]
- Subtract this from the original dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
3. Second term of the quotient:
- 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].
- Multiply [tex]\(-11\)[/tex] by the entire divisor:
[tex]\[
-11 \times (x^2 + 3x + 3) = -11x^2 - 33x - 33
\][/tex]
- Subtract this from the current remainder:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
4. Result:
- The remainder after the polynomial division is [tex]\(28x + 30\)[/tex].
Thus, when [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by [tex]\(x^2 + 3x + 3\)[/tex], the remainder is [tex]\(28x + 30\)[/tex].
1. Setup the Division:
- We have the dividend as [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex].
- The divisor is [tex]\(x^2 + 3x + 3\)[/tex].
2. Division Process:
- First term of the quotient: 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].
- Multiply [tex]\(3x\)[/tex] by the entire divisor [tex]\((x^2 + 3x + 3)\)[/tex]:
[tex]\[
3x \times (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x
\][/tex]
- Subtract this from the original dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
3. Second term of the quotient:
- 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].
- Multiply [tex]\(-11\)[/tex] by the entire divisor:
[tex]\[
-11 \times (x^2 + 3x + 3) = -11x^2 - 33x - 33
\][/tex]
- Subtract this from the current remainder:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
4. Result:
- The remainder after the polynomial division is [tex]\(28x + 30\)[/tex].
Thus, when [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by [tex]\(x^2 + 3x + 3\)[/tex], the remainder is [tex]\(28x + 30\)[/tex].
