



It is
not always so easy to find the places



where
the graph line crosses the X axis. 


Those
are the places where Y = 0. 


It
would be great if there was a way to figure out 


where
these places were. 





WE
CAN! 


Sometimes ... 





Say
we have some nasty equation like: 





Y
= 2X^{3}  3X^{2}  2X + 3






Does
this equation cross the X axis? 


WHO
KNOWS??? 





We
can guess at X values that MIGHT make Y = 0. 


We
can graph the equation and hope we can tell 


where
the line crosses the X axis. 


OR
... 


We
can use a new trick. 





For
this trick to work, all of the coefficients (the numbers) 


must
be integers. 


Integers
are stuff like 6 and 2, and 0. 


Things
that are not integers are things like ^{1}/2 and 6.2 and p. 





So
since all of the numbers in this equation ARE integers, 


we
can use the new trick. 


Here
goes ... 





First
we build a fraction. 


Take
the term that has no X and divide it into prime factors. 


In
this problem, the term with no X is a +3. 


The
prime factors of +3 are: 





3
= 1 × 3
so we have a 1 and a 3






These
will be the numerators (numbers on top of a fraction), 


for
this trick. 





Now
take the coefficient of the term with the biggest exponent. 


In
this problem, the term with the biggest exponent is 2X
^{3} 


so
we want the 2. 


Divide
it into prime factors ... 





2
= 1 × 2
so we have a 1 and a 2






These
will be the denominators (numbers on the bottom of a fraction) 


for
our trick. 





Our
numerators are 1 and 3 


Our
denominators are 1 and 2 


List
all of the fractions that have 1 or 3 on top 


and
1 or 2 on the bottom. 











If
the equation crosses the X axis at any rational number, 


it
will happen at one or more of these places: 











Great,
now what? 


Now
we use synthetic division. 


When
the equation crosses the X axis, Y = 0. 


That
means the remainder we get with synthetic division 


will
be ZERO! 





Most
people don't use synthetic division very much 


so
take a minute to look up how to do it if you want. 





Back
already? OK let's do it. 





Check
the 3: 









Check
some more numbers: 









So
the equation line crosses the X axis at X = 1 


Let's
see if it crosses any where else that we can find ... 












S0
the equation line also crosses the X axis at X = 1. 


Now
we get to so the hard ones ... 




















So
the equation line crosses the X axis at 


X
= 1, X = 1, and X = ^{3}/2 





Remember
that to use this trick 


all
of the numbers in the equations MUST be integers, 


but
the numbers we get for X don't have to be integers. 





There
is another cute trick that we can use. 


Since
X
= 1, X = 1, and X = ^{3}/2 are all values that make Y = 0. 


(X
 (1)), (X  1), and (X  ^{3}/2) are all factors of Y = 2X
^{3}
 3X ^{2} 2X + 3 





Are
they ALL the factors of the equation? 


Let's
find out. 


Multiply
them all together and see what you get ... 








That's
close but something is missing. 


We
want: 


2X^{3}  3X^{2}  2X + 3



and
we have: 








See
what's missing? 


We
need to multiply each term by 2. 


So
that's the missing factor, a 2! 





That
means: 


Y
= 2X^{3}  3X^{2}  2X + 3



Factors
as: 








And
the ONLY RATIONAL NUMBER places that the equation line 


crosses
the X axis are: 








You
keep saying "RATIONAL NUMBERS," why? 





Because
the graph line could cross the X axis 


at
an irrational number like p. 


If
it does, this trick won't help us find it. 





copyright 2005 Bruce Kirkpatrick 
