Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.
f(x)=x^2−3x+2 / x^10−x^9

Verified step by step guidance

1. Step 1: Identify the function's denominator, which is \( x^{10} - x^9 \). Set the denominator equal to zero to find the values of \( x \) that make the function undefined: \( x^{10} - x^9 = 0 \).
2. Step 2: Factor the denominator. Notice that \( x^{10} - x^9 \) can be factored as \( x^9(x - 1) = 0 \).
3. Step 3: Solve the factored equation \( x^9(x - 1) = 0 \) to find the critical points. This gives \( x^9 = 0 \) and \( x - 1 = 0 \).
4. Step 4: Solve \( x^9 = 0 \) to find \( x = 0 \). Solve \( x - 1 = 0 \) to find \( x = 1 \). These are the potential vertical asymptotes.
5. Step 5: Verify if these points are indeed vertical asymptotes by checking the behavior of the function as \( x \) approaches these values. Since the numerator \( x^2 - 3x + 2 \) does not have \( x = 0 \) or \( x = 1 \) as roots, both \( x = 0 \) and \( x = 1 \) are vertical asymptotes.