What are the additional linear factors of x^3−6x^2+11x−6 if x−3 is a factor? (1 point)
(x+2)(x+1)
(x−2)(x−1)
(x+2)(x−1)
(x−2)(x+1)
(x+2)(x-1)
To find the additional linear factors of a polynomial if one factor is known, we can perform polynomial long division by dividing the polynomial by the known factor.
In this case, we want to find the additional factors of x^3 − 6x^2 + 11x − 6 if x − 3 is a factor.
Performing the polynomial long division, we get:
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x - 3 | x^3 - 6x^2 + 11x - 6
- (x^3 - 3x^2)
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-3x^2 + 11x
- (-3x^2 + 9x)
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2x - 6
- (2x - 6)
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0
Since the remainder is 0, we can conclude that x - 3 is indeed a factor of x^3 - 6x^2 + 11x - 6.
To find the other factors, we can look at the remaining quotient, which is 2x - 6.
Setting it equal to 0 and solving for x, we get:
2x - 6 = 0
2x = 6
x = 3
So, the additional linear factor is x - 3.
Therefore, the correct answer is (x - 3)(x + 2)(x - 1).