The Rational Zero Theorem is a theorem that allows one to find out what the roots actually are in an equation. It does not state exactly what they are, but limits the possibilities on what the real (positive and negative) roots can be. Look at an example. In the function f(x) = 2x⁵ +9x⁴ +4x³ -x² +7x + 3 what is important with the Rational Zero Theorem is the first coefficient and the last number. The first coefficient is the 2 in front of the x⁵, the last number is the 3. These are the only numbers that are of interest at this point. Use the 2 and the 3 to determine what the possible real roots are. Take all the factors, numbers that go into it, of the 3 and put them over the factors of 2. All the factors of 3 are 3 and 1, and all the factors of 2 are 2 and 1. That means that by putting the factors of 3 over the factors of 2, one would get 4 different possibilities.

Always simplify these, so in this case they are . Next put a positive and negative in front of each number.

 

. This produces eight different numbers, .

 

 

What this says is that these are the only possible real values, or otherwise called rational zeros of the function f(x) = 2x⁵ +9x⁴ +4x³ -x² +7x + 3. Later on it will be discussed how to find out which ones are the actual zero’s of the function.

 

 

 

Example

 

State all the possible rational zero’s of the function f(x) = x³ -x² +7x + 5

 

Put all the factors of 5 (the number by itself) over all the factors of 1, the leading coefficient.

 

 

Add the positives and negatives and simplify.

 

±5,±1

 

It is generally required to write these from smallest to largest.

 

±1,±5

 

Answer!

 

 

 

 

 

 

Example

State all the possible rational zero’s of the function f(x) = x⁴ +3x³ -5x² +2x – 18

 

The leading coefficient is 1 and the only factor of 1 is 1. The last number, 18, has a lot of factors. They are 1,2,3,6,9, and 18. So take each of these and put them each over 1.

 

±1,±2,±3,±6,±9,±18

 

Answer!

 

 

 

 

Example

 

Find all the possible rational zero’s of f(x) = 2x³ +5x² +6x – 8

 

Put all the factors of 8 over all the factors of 2.

 

 

Simplify these.

 

 

The 1,2,and 4 are repeated. There is no reason to write these twice. Rewrite in order and put a ± in front of each one.

 

 

Answer!

 

 

 

 

 

Example

 

Find all the possible rational zero’s of f(x) = 6x³ +5x² -3x + 15

 

Put all the factors of 15(1,3,5,and 15) over all the factors of 6(1,2,3, and 6).

 

 

Simplify each one.

 

 

 

Put a ± sign in front of each one and list from smallest to greatest.

 

 

Answer! These are all the different possible rational zero’s of this function.