Expert: Paul Klarreich - 3/6/2007

Question
Hello Paul,
Although I thought this was easy I am getting no where with it... it is simple,but I just cannot figure out how to do it!

The x intercepts in exact values of the following equation
f:R->R, f(x)=(x-1)(x-3)(x+2)+4

The previous question involved addition of ordinates so I am not sure if it has anything to do with that or not. Thank you very much for your help, Belinda.  

Answer
Questioner:   Belinda
Category:  Advanced Math
 
Subject:  x intercepts
Question:  Hello Paul,
Although I thought this was easy I am getting no where with it... it is simple,but I just cannot figure out how to do it!

The x intercepts in exact values of the following equation
f:R->R, f(x)=(x-1)(x-3)(x+2)+4

The previous question involved addition of ordinates so I am not sure if it has anything to do with that or not. Thank you very much for your help, Belinda.
...................................
Hi, Belinda,

For your function:

f(x)=(x-1)(x-3)(x+2)+4

Start by multiplying out:

f(x) = (x^2 - 4x + 3)(x + 2) + 4

f(x) = x^3 - 4x^2 + 3x + 2x^2 - 8x + 6 + 4

f(x) = x^3 - 2x^2 - 5x + 10

Now look for rational roots.  The possibilities are:  +- 1,2,5,10
+-1 is not going to work.  Try 2:

1   - 2   - 5   10   (2)
     2     0  -10
------------------------
1     0   - 5    0

Aha!  x = 2 is a root, (an x-intercept) and the reduced function is

x^2 - 5 = 0, which has roots  x = +- sqrt(5)