Question:
Write all the points where $\cfrac{n^2-9n+20}{n^4-n^3-3n^2+n+2}$ is discontinuous.Provide the answer in simplest form.
My Approach:
Should I convert the numerator and denominator into its respective product form and try to cancel them? I am not getting any idea. Any help or guidance to solve this problem would be appreciated.
On
Hint:
You should try to find the product form: $(x-a_1)(x-a_2)\cdots (x-a_n)$.
My tip is that the $a_i$'s are exactly the zeros of the polynomials. You should know how to find the zeros of a quadratic equation. Also you can try some simple numbers for the more complex polynomial.
Example. The polynomial $x+x^2$ has the zeros $0$ and $-1$, hence the product form is $$(x-(-1))(x-0)=(x+1)x.$$
Try to apply this to your problem.
Note:$\;$ Sometimes the product form described above does not exactly equal the polynomial (if there is zero of multiplicity $>2$), but you should not care about this for now.
Use the Rational Root Theorem and synthetic division on the denominator. You will find 2 distinct factors with multiplicity 1 and a third with multiplicity 2.