Finding Holes
Factor the following rational functions:
a) \(f(x)=\dfrac{3x^2-12x-15}{x^2-3x-10}\)
b) \(f(x)=\dfrac{x^2-9}{x-3}\)
c) \(f(x)=\dfrac{4x^2+8x-12}{x^2-4x-21}\)
Once factored, note that binomials that are both in the numerator and the denominator can be divided out as a factors of one. Setting those factors equal to zero show you where the “holes” are in the function when it is graphed. For example, for function b), \(\dfrac{(x-3)}{(x-3)}\) can be divided out of the function. So there is a hole where \(x\) is equal to three. Evaluating for three in the simplified version of the function, the \(y\)-value would have been 6. So, the hole is at \((3,6)\) on the graph.
