To verify if [tex]\( g(x) = x - 2 \)[/tex] is a factor of [tex]\( f(x) = x^3 - 3x^2 + 4x - 4 \)[/tex], we can perform polynomial division of [tex]\( f(x) \)[/tex] by [tex]\( g(x) \)[/tex] and check the remainder. If the remainder is zero, then [tex]\( g(x) \)[/tex] is a factor of [tex]\( f(x) \)[/tex].
### Step-by-Step Solution:
1. Set up the division: We need to divide [tex]\( f(x) \)[/tex] by [tex]\( g(x) \)[/tex]:
[tex]\( f(x) = x^3 - 3x^2 + 4x - 4 \)[/tex]
[tex]\( g(x) = x - 2 \)[/tex]
2. Perform the polynomial division:
- First term: Divide the leading term of [tex]\( f(x) \)[/tex] by the leading term of [tex]\( g(x) \)[/tex]:
[tex]\[
\frac{x^3}{x} = x^2
\][/tex]
- Multiply [tex]\( g(x) \)[/tex] by [tex]\( x^2 \)[/tex] and subtract from [tex]\( f(x) \)[/tex]:
[tex]\[
(x^3 - 3x^2 + 4x - 4) - (x^2 \cdot (x - 2)) = (x^3 - 3x^2 + 4x - 4) - (x^3 - 2x^2) = -x^2 + 4x - 4
\][/tex]
- Second term: Divide the new leading term by the leading term of [tex]\( g(x) \)[/tex]:
[tex]\[
\frac{-x^2}{x} = -x
\][/tex]
- Multiply [tex]\( g(x) \)[/tex] by [tex]\(-x\)[/tex] and subtract from the new polynomial:
[tex]\[
(-x^2 + 4x - 4) - (-x \cdot (x - 2)) = (-x^2 + 4x - 4) - (-x^2 + 2x) = 2x - 4
\][/tex]
- Third term: Divide the new leading term by the leading term of [tex]\( g(x) \)[/tex]:
[tex]\[
\frac{2x}{x} = 2
\][/tex]
- Multiply [tex]\( g(x) \)[/tex] by 2 and subtract from the new polynomial:
[tex]\[
(2x - 4) - (2 \cdot (x - 2)) = (2x - 4) - (2x - 4) = 0
\][/tex]
3. Conclusion:
After performing the division, we have the quotient [tex]\( x^2 - x + 2 \)[/tex] and the remainder [tex]\( 0 \)[/tex].
Since the remainder is zero, [tex]\( g(x) = x - 2 \)[/tex] is indeed a factor of [tex]\( f(x) = x^3 - 3x^2 + 4x - 4 \)[/tex].
Thus, we have verified that [tex]\( g(x) \)[/tex] is a factor of [tex]\( f(x) \)[/tex]. The quotient is [tex]\( x^2 - x + 2 \)[/tex] and the remainder is [tex]\( 0 \)[/tex].
