Which polynomial is a factor of $x^3 - 5x^2 + 11x - 10$? A - HSC - SSCE Mathematics Extension 1 - Question 1 - 2017 - Paper 1

To determine which polynomial is a factor of the given polynomial, we need to test each option by using polynomial division or the factor theorem. The factor theorem states that for a polynomial $f(x)$, if $f(c) = 0$, then $(x - c)$ is a factor of $f(x)$.

Let's evaluate the options:

1. Option A: $x - 2$
   Substitute $x = 2$ into the polynomial: $f(2) = 2^3 - 5(2^2) + 11(2) - 10$
   $= 8 - 20 + 22 - 10 = 0$
   Hence, $x - 2$ is a factor.

2. Option B: $x + 2$
   Substitute $x = -2$ into the polynomial: $f(-2) = (-2)^3 - 5(-2)^2 + 11(-2) - 10$\

   $$$$

eq 0$$
Thus, $x + 2$ is not a factor.

3. Option C: $11x - 10$
   This is not in the form of $(x - c)$. We do not test this directly.

4. Option D: $x^2 - 5x + 11$
   This is again not in a simple linear factor form; we'll skip its evaluation as we already found a valid factor.

Since in our evaluation, only option A yielded a value of zero, we have concluded that the correct answer is A: $x - 2$.