Expert: Paul Klarreich - 7/2/2009

Question
Hi Paul Klarreich,

I have been some trouble lately finding with my precalulus-
it is just confusing..please can you help me ?

Find the x and y intercepts of each polynomial function
and determine whether the graph of f touches or crosses
the x axis at each x intercept..

1. f(x) = x(x-2) (x-4)

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

3. f(x) = x^3 + 4x

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

This might seem like a lot of problems...but i  really need
some help on these..any help would be greatly appreciated.
Thanks !

Answer
Questioner:   Rajah
Country:  United States
Category:  Advanced Math
Private:  No
 
Subject:  Pre calculus help...
Question:  Hi Paul Klarreich,

I have been some trouble lately finding with my precalulus-
it is just confusing..please can you help me ?

Find the x and y intercepts of each polynomial function
and determine whether the graph of f touches or crosses
the x axis at each x intercept..

1. f(x) = x(x-2) (x-4)

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

3. f(x) = x^3 + 4x

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

This might seem like a lot of problems...but i  really need
some help on these..any help would be greatly appreciated.
Thanks !
.......................................
Hi, Rajah,

Here are the clues:

1. If you have  (x - a) as a factor, then  x = a  is an intercept.
2. A factor of  x is a factor of (x - 0)

3A. If (x - a) has a multiplicity (the exponent) that is even, then it touches.
3B. If (x - a) has a multiplicity (the exponent) that is odd,  then it crosses.

Your examples:

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

x = 2 has multiplicity 1 (odd).  Crosses at x = 2.
x = -4 has multiplicity 2 (even).  Touches at x = -4.

I think you can do the others now.