The Rational Root Theory

FST Rational Root Notes

Learn about Rational Root Theorem to find rational & irrational zeros especially when the polynomial is not factorable.

Rational Zeros

• 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, . ..)

Rational Zero Theorem

•  $If\ f\left(x\right)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ 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 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 

Multiple Choice

2) List all possible rational zeros of  $$f\left(x\right)=x^5+9x^3-2x^2-18$$  
Possible rational roots are $$\frac{p}{q}=\frac{factors\ of\ cons\tan t\ term}{factors\ of\ leading\ coefficients}$$  

• $$\pm1,\pm2,\pm3$$  

• $$\pm1,\ \pm2,\ \pm3,\ \pm6$$  
  

• $$\pm1,\ \pm2,\ \pm3,\pm6,\ \pm9,\pm18$$  
  

Multiple Choice

3) List all possible rational zeros of  $$f\left(x\right)=4x^3-4x^2-x+1$$  
Possible rational roots are $$\frac{p}{q}=\frac{factors\ of\ cons\tan t\ term}{factors\ of\ leading\ coefficients}$$  

• $$\pm1,\pm2,\pm4$$  

• $$\pm1,\ \pm\frac{1}{2},\ \pm\frac{1}{4}$$  
  

• $$\pm1,\ \pm2,\ \pm4,\pm8$$  
  

• $$\pm1,\pm4$$  

Multiple Choice

4) List all possible rational zeros of  $$f\left(x\right)=3x^3-x^2-18x+16$$  
Possible rational roots are $$\frac{p}{q}=\frac{factors\ of\ cons\tan t\ term}{factors\ of\ leading\ coefficients}$$  

• $$\pm1,\pm\frac{1}{3},\pm2,\pm\frac{2}{3},\pm4,\pm\frac{4}{3},\pm8,\pm\frac{8}{3},\pm16,\pm\frac{16}{3}$$  

• $$\pm1,\ \pm\frac{1}{3},\ \pm\frac{16}{3}$$  
  

• $$\pm1,\ \pm2,\ \pm4,\pm8,\pm16$$  
  

Finding Rational Zeros

• 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.

Multiple Select

7) List all possible rational zeros of  $$f\left(x\right)=x^3+3x^2-6x-8$$  Then, find the actual zeros.

• $$\pm1,\pm\frac{1}{8}$$  
  

• $$\pm1,\ \pm\frac{1}{3},\ \pm\frac{16}{3}$$  
  

• $$\pm1,\ \pm2,\ \pm4,\pm8$$  
  

• x = -1

• x = 2, -4

Multiple Select

8) List all possible rational zeros of  $$f\left(x\right)=x^3+5x^2-25x+19$$  Then, find the all the real zeros. (4th page of notes).

• $$\pm1,\pm19$$  
  

• $$\pm1,\ \pm\frac{1}{19},\ \pm19$$  
  

• $$x\ =\ \pm2\sqrt{7}$$  
  

• $$x\ =\ 1$$  
  

• $$x=-3\pm2\sqrt{7}$$  
  

Poll

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Open Ended

Write a summary about Rational Root Theorem, using lower bound & upper bound to find all real zeros.