What are the additional linear factors of x3−6x2+11x−6 if x−3 is a factor? (1 point)

Question

Responses

(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

(x−2)(x−1)

Answer 1

To determine the additional linear factors of the polynomial \( x^3 - 6x^2 + 11x - 6 \) given that \( x - 3 \) is a factor, we can perform polynomial division to find the quotient when dividing by \( x - 3 \).

1. Perform synthetic division of the polynomial \( x^3 - 6x^2 + 11x - 6 \) by \( x - 3 \):

   • Coefficients of \( x^3 - 6x^2 + 11x - 6 \): \( 1, -6, 11, -6 \)
   • Root from \( x - 3 = 0 \): \( 3 \)

   Synthetic Division Steps:

   3 |  1  -6   11  -6
     |      3   -9   6
   ---------------------
       1  -3    2   0
   

   The result of the synthetic division gives us the quotient \( x^2 - 3x + 2 \).

2. Factor the quotient \( x^2 - 3x + 2 \):

   • We can factor it as follows:

   \[ x^2 - 3x + 2 = (x - 1)(x - 2) \]

3. Combine the factors: The original polynomial \( x^3 - 6x^2 + 11x - 6 \) can be expressed as:

   \[ (x - 3)(x - 1)(x - 2) \]

Thus, the additional linear factors are \((x - 2)(x - 1)\).

Among your response options, the correct answer is:

(x−2)(x−1).