Prove the Rational Zeros Theorem.

Let p/q, where p and q have no common factors except 1 and -1, be a zero of the polynomial function

 

whose coefficients are all integers. Show that

Now, because p is a factor of the first n terms of this equation, p must also be a factor of the term a₀qⁿ. Since p is not a factor of q (why?), p must be a factor of a₀. Similarly, q must be a factor of a_n.

anx" + an-1x"-1 |f(x) = a + ajx + ao + do ap" + an-1p"-'q+ + ajpq"- + aoq" = 0