Unit G SUmmary Question # 9: X-Intercepts of a Rational Function

9. Describe how to find the x-intercepts of a rational function. Include both the long way and the shortcut way, explaining why the shortcut makes mathematical sense.

How to: The LOOOOONG WAY....
In order to find the x-intercepts through the use of the faster method, it is important that you know how we got to that point and why.
To make it easier, lets use an example: 3x+3/x^2-4
     ^* REMEBER TO FACTOR!--->  3(x+1)/(x+2)(x-2)
       * now we get to the fun part..... x-intercepts!!! :D
       First: set the y [or f(x) ] to zero, thus setting the equation to zero
                                    ^   0= 3(x+1)/(x+2)(x-2)
     
       Second: Mulitply both sides by the denominator
                                  ^   (x+2)(x-2) 0=3(x+1)/(x+2)(x-2)    x    (x+2)(x-2)
     
       Third: As you can see, the denominator cancels out on the right side
                                 ^ (x+2)(x-2) 0=3(x+1)/(x+2)(x-2)    x    (x+2)(x-2)

                                               * now you are left with ....
                                                   (x+2)(x-2) 0=3(x+1)
       
         Fourth: as you can see, the (x+2)(x-2) will be eliminate becauseANYTHING                 multiplied by zeor will turn to...... zero!

         Fifth: Now you have:
                                                          0=3(x+1)
     
          Sixth: divide by 3:
                                                          0=3(x+1)
                                                          3       3

       Seventh: solve like you normally would.... in this case subtract 1 to both sides
                                                          0-1=x+1 -1
 
      Eighth!!! And your x-intercept is....      x=-1  making it (-1,0)


How to: SHORT WAY...
Phew! Now that you know how to do it the long way, the shorcut will make a lot of sense! :)

          First: set the y [or f(x) ] to zero, thus setting the equation to zero

                                             ^   0= 3(x+1)/(x+2)(x-2)

         Second: since you know that the denominator will always cancel and be eliminated you can keep on moving forward. SIGH OF RELIEF....

                                                      0= 3(x+1)/(x+2)(x-2)

         Third: You are left with...

                                                       0=3(x+1)

       

          Fourth: divide by 3:

                                                          0=3(x+1)

                                                          3       3

      Fifth: solve like you normally would.... in this case subtract 1 to both sides

                                                          0-1=x+1 -1

   

      Sixth!!! And your x-intercept is....      x=-1  making it (-1,0)