To find the additional linear factors of the polynomial \( f(x) = x^3 - 6x^2 + 11x - 6 \) given that \( x - 3 \) is a factor, we can use polynomial long division to divide \( f(x) \) by \( x - 3 \).
- Set up the long division of \( f(x) \) by \( x - 3 \).
Perform the division:
- Divide \( x^3 \) by \( x \), which gives \( x^2 \).
- Multiply \( x^2 \) by \( x - 3 \): \[ x^2(x - 3) = x^3 - 3x^2 \]
- Subtract this from \( f(x) \): \[ (x^3 - 6x^2 + 11x - 6) - (x^3 - 3x^2) = -6x^2 + 3x^2 + 11x - 6 = -3x^2 + 11x - 6 \]
- Now, divide \( -3x^2 \) by \( x \), which gives \( -3x \).
- Multiply \( -3x \) by \( x - 3 \): \[ -3x(x - 3) = -3x^2 + 9x \]
- Subtract: \[ (-3x^2 + 11x - 6) - (-3x^2 + 9x) = 11x - 9x - 6 = 2x - 6 \]
- Now, divide \( 2x \) by \( x \), which gives \( 2 \).
- Multiply \( 2 \) by \( x - 3 \): \[ 2(x - 3) = 2x - 6 \]
- Subtract: \[ (2x - 6) - (2x - 6) = 0 \]
So, the division results in: \[ f(x) = (x - 3)(x^2 - 3x + 2) \]
- Now we need to factor \( x^2 - 3x + 2 \): \[ x^2 - 3x + 2 = (x - 1)(x - 2) \]
Thus, the complete factorization of the polynomial \( f(x) \) is: \[ f(x) = (x - 3)(x - 1)(x - 2) \]
The additional linear factors are \( x - 1 \) and \( x - 2 \).