What are the additional linear factors of x3−6x2+11x−6 if x−3 is a factor? (1 point)
Responses
(x−2)(x−1)
left parenthesis x minus 2 right parenthesis left parenthesis x minus 1 right parenthesis
(x+2)(x−1)
left parenthesis x plus 2 right parenthesis left parenthesis x minus 1 right parenthesis
(x−2)(x+1)
left parenthesis x minus 2 right parenthesis left parenthesis x plus 1 right parenthesis
(x+2)(x+1)
left parenthesis x plus 2 right parenthesis left parenthesis x plus 1 right parenthesis
The correct answer is: (x+2)(x-1) or left parenthesis x plus 2 right parenthesis left parenthesis x minus 1 right parenthesis
To find the additional linear factors of the given polynomial when x-3 is a factor, we can use long division or synthetic division to divide the polynomial by x-3.
Using long division:
- Set up the long division with the given polynomial (x^3 - 6x^2 + 11x - 6) as the dividend and x-3 as the divisor.
- Divide the first term (x^3) by x, which gives x^2 as the quotient.
- Multiply the divisor (x-3) by the quotient (x^2) and subtract it from the dividend (x^3 - 6x^2 + 11x - 6).
- Continue the long division process until no more terms can be divided evenly.
The result of the long division is:
x^2 - 3x + 2
This means that (x-3) divided evenly into the given polynomial, and the remaining factor is (x^2 - 3x + 2).
Now, we can factor the remaining quadratic factor (x^2 - 3x + 2) to find the additional linear factors. Factoring the quadratic, we get:
(x-2)(x-1)
Therefore, the additional linear factors of x^3 - 6x^2 + 11x - 6 when x-3 is a factor are (x-2) and (x-1).
So, the correct response is: (x-2)(x-1)