Sure, let's solve the problem of finding the remainder when [tex]\(x^3 - 3x^2 + 4x - 7\)[/tex] is divided by [tex]\(x + 2\)[/tex].
We will use polynomial division to solve this problem.
1. Setup the division:
- Dividend: [tex]\( x^3 - 3x^2 + 4x - 7 \)[/tex]
- Divisor: [tex]\( x + 2 \)[/tex]
2. Divide the first term of the dividend by the first term of the divisor:
- [tex]\( \frac{x^3}{x} = x^2 \)[/tex]
3. Multiply [tex]\( x^2 \)[/tex] by the entire divisor [tex]\( x + 2 \)[/tex]:
- [tex]\( x^2 \cdot (x + 2) = x^3 + 2x^2 \)[/tex]
4. Subtract this from the original polynomial to get the new polynomial:
- [tex]\((x^3 - 3x^2 + 4x - 7) - (x^3 + 2x^2) = -5x^2 + 4x - 7\)[/tex]
5. Repeat this process with the new polynomial:
- Divide the first term of the new polynomial by the first term of the divisor:
- [tex]\( \frac{-5x^2}{x} = -5x \)[/tex]
- Multiply [tex]\(-5x\)[/tex] by [tex]\( x + 2 \)[/tex]:
- [tex]\( -5x \cdot (x + 2) = -5x^2 - 10x \)[/tex]
- Subtract this from the new polynomial:
- [tex]\((-5x^2 + 4x - 7) - (-5x^2 - 10x) = 14x - 7\)[/tex]
6. Repeat the division process again:
- Divide the first term of the new polynomial by the first term of the divisor:
- [tex]\( \frac{14x}{x} = 14 \)[/tex]
- Multiply [tex]\( 14 \)[/tex] by [tex]\( x + 2 \)[/tex]:
- [tex]\( 14 \cdot (x + 2) = 14x + 28 \)[/tex]
- Subtract this from the polynomial:
- [tex]\( (14x - 7) - (14x + 28) = -35 \)[/tex]
7. The final remainder after subtracting and division is [tex]\(\boxed{-35}\)[/tex].
Thus, the remainder when [tex]\( x^3 - 3x^2 + 4x - 7 \)[/tex] is divided by [tex]\( x + 2 \)[/tex] is [tex]\( \boxed{-35} \)[/tex]. The correct answer is [tex]\( \text{C. -35} \)[/tex].
