Find polynomials p and q such that f=p/q is undefined at 1 and 2, but f has a vertical asymptote only at 2. Sketch a graph of your function.

Verified step by step guidance

1. Step 1: Understand the problem requirements. We need to find polynomials p(x) and q(x) such that the rational function f(x) = \frac{p(x)}{q(x)} is undefined at x = 1 and x = 2, but has a vertical asymptote only at x = 2.
2. Step 2: Recall that a rational function is undefined where its denominator is zero. To make f(x) undefined at x = 1 and x = 2, q(x) should have factors (x - 1) and (x - 2).
3. Step 3: To ensure a vertical asymptote at x = 2, the factor (x - 2) should not be canceled out by the numerator p(x). Therefore, p(x) should not have the factor (x - 2).
4. Step 4: To make f(x) defined at x = 1, the factor (x - 1) in q(x) should be canceled by a similar factor in p(x). Thus, p(x) should have the factor (x - 1).
5. Step 5: Construct the polynomials. Let p(x) = (x - 1) and q(x) = (x - 1)(x - 2). This ensures f(x) = \frac{(x - 1)}{(x - 1)(x - 2)} is undefined at x = 1 and 2, but has a vertical asymptote only at x = 2.