Rational Root Theorem Lesson

Recall that a polynomial function of degree n can have at most n real zeros.

Real zeros can be rational or irrational.

Rational zeros are those that can be written in the form of a fraction p/q using integers (. . . -3, -2, -1, 0, 1, 2, 3, . ..)

 $If\ f\left(x\right)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ has integer coefficients, then every rational zero of the function has the following term:

 $\frac{p}{q}=\frac{factors\ of\ cons\tan t\ p}{factors\ of\ leading\ coefficient\ q}=possible\ rational\ zeros$  

List all possible rational zeros of each function.

1)   $f\left(x\right)=x^4+2x^2-x+2$ 

Step 1) Find  $\frac{cons\tan t}{leading\ coefficient}$  ;  $\frac{2}{1}$  

Step 2) List factors with  $\pm$  ;  $\frac{\pm1,\pm2}{\pm1}$  

Step 3) Simplify; ±1, ±2 

Therefore, possible rational zeros are ±1, ±2 

Step 1) List all possible rational zeros of each function using the Rational Root Theorem

Step 2) Test the zeros using synthetic substitution

Step 3) When you find a zero that works, use the remainder to completely factor the polynomial. Then find zeros.