Answer :
To find the remainder when [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by [tex]\(x^2 + 3x + 3\)[/tex], we use polynomial division. Here's how the process works:
1. Identify the Dividend and Divisor:
- Dividend: [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex]
- Divisor: [tex]\(x^2 + 3x + 3\)[/tex]
2. Perform Polynomial Long Division:
- Divide the leading term of the dividend by the leading term of the divisor. Here, divide [tex]\(3x^3\)[/tex] by [tex]\(x^2\)[/tex], which gives [tex]\(3x\)[/tex].
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex], resulting in [tex]\(3x^3 + 9x^2 + 9x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x)
\][/tex]
This simplifies to: [tex]\(-11x^2 - 5x - 3\)[/tex].
3. Repeat the Process:
- Now divide the new leading term [tex]\(-11x^2\)[/tex] by the leading term [tex]\(x^2\)[/tex], which gives [tex]\(-11\)[/tex].
- Multiply the entire divisor by [tex]\(-11\)[/tex], resulting in [tex]\(-11x^2 - 33x - 33\)[/tex].
- Subtract this from [tex]\(-11x^2 - 5x - 3\)[/tex]:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33)
\][/tex]
This simplifies to: [tex]\(28x + 30\)[/tex].
4. Determine the Remainder:
- Once the degree of what's left (which is [tex]\(28x + 30\)[/tex]) is less than the degree of the divisor ([tex]\(x^2 + 3x + 3\)[/tex]), this is 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].
Therefore, the correct answer is [tex]\( \boxed{28x + 30} \)[/tex].
1. Identify the Dividend and Divisor:
- Dividend: [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex]
- Divisor: [tex]\(x^2 + 3x + 3\)[/tex]
2. Perform Polynomial Long Division:
- Divide the leading term of the dividend by the leading term of the divisor. Here, divide [tex]\(3x^3\)[/tex] by [tex]\(x^2\)[/tex], which gives [tex]\(3x\)[/tex].
- Multiply the entire divisor [tex]\(x^2 + 3x + 3\)[/tex] by [tex]\(3x\)[/tex], resulting in [tex]\(3x^3 + 9x^2 + 9x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x)
\][/tex]
This simplifies to: [tex]\(-11x^2 - 5x - 3\)[/tex].
3. Repeat the Process:
- Now divide the new leading term [tex]\(-11x^2\)[/tex] by the leading term [tex]\(x^2\)[/tex], which gives [tex]\(-11\)[/tex].
- Multiply the entire divisor by [tex]\(-11\)[/tex], resulting in [tex]\(-11x^2 - 33x - 33\)[/tex].
- Subtract this from [tex]\(-11x^2 - 5x - 3\)[/tex]:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33)
\][/tex]
This simplifies to: [tex]\(28x + 30\)[/tex].
4. Determine the Remainder:
- Once the degree of what's left (which is [tex]\(28x + 30\)[/tex]) is less than the degree of the divisor ([tex]\(x^2 + 3x + 3\)[/tex]), this is 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].
Therefore, the correct answer is [tex]\( \boxed{28x + 30} \)[/tex].
