Rational Zero Theorem

Introduction

If a polynomial f(x) with integer coefficients has a rational zero p/q in lowest terms, then p must be a factor of the constant term and q must be a factor of the leading coefficient

Put into English by the robot

If a polynomial has a rational root, the only possible candidates come from:
- top = factors of the constant term
- bottom = factors of the leading coefficient

A rational root is a solution that is a normal fraction — like 3, −2, 5/3, −7/4, etc.
Not a square root, not an imaginary number, not something messy.
Just a clean fraction.

Example 1

So started off with an example. The important thing was p and q, p being the coefficient of the highest power and q being the constant. So given

f(x) = x³ +2x² -5x - 6, q = 1, p = 6

Now we list the possible factore of p and q

 p   +/-1, +/-2, +/-3, +/-6, 
 - = -----------------------
 q             +/-1     

Any number divided by 1 is itself so the possible rational zeros are

   = +/-1, +/-2, +/-3, +/-6

So it we set the function to 0

   x³ +2x² -5x - 6, q = 1, p = 6 = 0

Then the solutions to this equation be some of these numbers listed

They proceeded to test which numbers working starting a 1,

 = 1³ +2(1)² -5(1) - 6
 = 1 + 2 -5 -6
 = -8 So not 1

They proceeded to test which numbers working starting a 2,

 = 2³ +2(2)² -5(2) - 6
 = 8 + 8 - 10 - 6
 = 16 -16 = 0

So we have found one. Once we have found one we can so synthetic division to find another.

2 | 1 + 2  -5 -6
  | ↓   2   8  6
   -------------
    1  +4  +3  0

    1x²+4x +3 = 0 (factoring the expression gives)
 
   (x + 3)(x + 1) = 0