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 perform polynomial division.
### Step-by-Step Solution:
1. Set Up the Division:
We need to divide the polynomial [tex]\( 3x^3 - 2x^2 + 4x - 3 \)[/tex] (the dividend) by [tex]\( x^2 + 3x + 3 \)[/tex] (the divisor).
2. Determine the First Term of the Quotient:
- Look at the leading term of the dividend: [tex]\( 3x^3 \)[/tex].
- Divide it by the leading term of the divisor: [tex]\( x^2 \)[/tex].
- This gives us [tex]\(\frac{3x^3}{x^2} = 3x\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor by the term found in step 2: [tex]\(3x \cdot (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x\)[/tex].
- Subtract this result from the dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the Division Process:
- Look at the new leading term: [tex]\(-11x^2\)[/tex].
- Divide it by the leading term of the divisor: [tex]\( x^2 \)[/tex].
- This gives [tex]\(\frac{-11x^2}{x^2} = -11\)[/tex].
5. Multiply and Subtract Again:
- Multiply the entire divisor by this new term: [tex]\(-11 \cdot (x^2 + 3x + 3) = -11x^2 - 33x - 33\)[/tex].
- Subtract this from the result of the previous subtraction:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
6. Identify the Remainder:
At this point, the remainder is [tex]\( 28x + 30 \)[/tex], which cannot be divided further by [tex]\( x^2 + 3x + 3 \)[/tex] because its degree is lower than that of the divisor.
Therefore, 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].
### Step-by-Step Solution:
1. Set Up the Division:
We need to divide the polynomial [tex]\( 3x^3 - 2x^2 + 4x - 3 \)[/tex] (the dividend) by [tex]\( x^2 + 3x + 3 \)[/tex] (the divisor).
2. Determine the First Term of the Quotient:
- Look at the leading term of the dividend: [tex]\( 3x^3 \)[/tex].
- Divide it by the leading term of the divisor: [tex]\( x^2 \)[/tex].
- This gives us [tex]\(\frac{3x^3}{x^2} = 3x\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor by the term found in step 2: [tex]\(3x \cdot (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x\)[/tex].
- Subtract this result from the dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the Division Process:
- Look at the new leading term: [tex]\(-11x^2\)[/tex].
- Divide it by the leading term of the divisor: [tex]\( x^2 \)[/tex].
- This gives [tex]\(\frac{-11x^2}{x^2} = -11\)[/tex].
5. Multiply and Subtract Again:
- Multiply the entire divisor by this new term: [tex]\(-11 \cdot (x^2 + 3x + 3) = -11x^2 - 33x - 33\)[/tex].
- Subtract this from the result of the previous subtraction:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
6. Identify the Remainder:
At this point, the remainder is [tex]\( 28x + 30 \)[/tex], which cannot be divided further by [tex]\( x^2 + 3x + 3 \)[/tex] because its degree is lower than that of the divisor.
Therefore, 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].
