given the rational function r(x)= (x^3+0.8x^2-19.55X+28.566) / (x^2-3.9x+3.68) find all the x values where the graph of r(x) has a hole.

Question

AP precalc

Answer 1

There is a hole (removable discontinuity) in the graph at x=c when x-c is a factor of the numerator and denominator which cancels out when the expressions are simplified and for which no x - c factors remain in the denominator after simplification.

Using the quadratic formula, the denominator is equal to zero when x = 2.3 or 1.6. So, the denominator factors as (x - 2.3)(x - 1.6).

By substituting 2.3 and 1.6 into the numerator, we see that 2.3 is a root of the numerator, but 1.6 is not.

So, x - 2.3 is also a factor of the numerator, but x - 1.6 is not. So, there is a hole in the graph when x = 2.3.

Answer 2

Holes will appear when the function is undefined.

In this case, that means the denominator is zero.

X^2 + 3.9x + 3.68 = 0

Using the quadratic formula, this equation holds when x = -1.6 or x = -2.3

This is where the holes are. (Removability of holes has not been examined since this was not asked for.)