



There
are some factors that show up all the time.



It
is a good idea to get to know these. 


They
are ... 





The
Perfect Squares: 


Using
"a" and "b" to stand for any old numbers we
might have, 


the
perfect squares are: 





(aX + b)^{2}
= a^{2}X^{2} + 2abX + b^{2}



(aX  b)^{2}
= a^{2}X^{2}  2abX + b^{2} 





When
these two are multiplied out like this 


they
are almost the same. 





The
only difference is that the one that starts with a minus 


has
a minus on the middle term. 





So
using real numbers and multiplying these out 


we
would get ... 





Examples: 






(X + 3)^{2}
= 1^{2}X^{2} + 2x1x3xX
+ 3^{2} = X^{2} + 6X + 9 







(3X + 5)^{2}
= 3^{2}X^{2} + 2x3x5xX
+ 5^{2} = 9X^{2} + 30X + 25 







(X^{2}  .5)^{2}
= .5^{2}X^{4}  2x1x.5xX^{2}
+ .5^{2} = .25X^{4}  X^{2} +
.25 





The
Perfect Cubes: 


Using
"a" and "b" to stand for any old numbers we
might have, 


the
perfect cubes are: 





(aX + b)^{3}
= a^{3}X^{3} + 3a^{2}bX^{2} + 3ab^{2}X
+ b^{3}



(aX + b)^{3}
= a^{3}X^{3}  3a^{2}bX^{2} + 3ab^{2}X
 b^{3} 





When
these two are multiplied out, 


the
only difference between them is also in the signs. 


The
signs on (aX + b) ^{3} are all positive 


The
signs on (aX  b) ^{3} alternate: positive  negative 
positive  negative 





So
using real numbers and multiplying these out 


we
would get ... 





Examples: 






(X + 3)^{3}
= 1^{3}X^{3} + 3x1^{2}x3xX^{2}
+ 3x1x3^{2}xX
+ 3^{3}
= X^{3}
+ 9X^{2} + 27X + 9 







(X  p)^{3}
= 1^{3}X^{3}  3x1^{2}xpxX^{2}
+ 3x1xp^{2}xX
 p^{3}
= X^{3}
+ 3pX^{2}
+ p^{2}X
+ p^{3} 





We
could keep on going with this stuff forever. 


We
could do (aX + b) ^{4}, (aX  b)_{7} and on and on, 


but
there is an easier way ... 





It's
called Pascal's triangle. 





It
starts like this. 


1
1






For
our purposes here, 


these
stand for the two terms in something you 


would
be multiplying times itself. 





Since
this is the start, 


it
shouldn't be too surprising that this stands for: 





(a + b)^{1}
which is just a + b






The
next step builds on the start above (the 1 1). 


It
looks like this ... 











The
following levels continue the pattern. 











OK,
this looks really cute. 


BUT
WHAT GOOD IS IT? 





Here's
the deal ... 











See
what happens? 


The
coefficients are the numbers from the triangle. 


Check
out what the exponents on X are doing. 


It's
a countdown ... 





We
could have a coefficient on X. 





Or
a number other than 1. 





To
deal with that, 


here's
the first 4 levels of the triangle 


with
"a" and "b" standing for any old numbers we
might have. 











Look
what happens. 


The
numbers in bold are the Pascal's triangle numbers. 


The
exponents on X go down 1 every term as we go from left to right. 


The
exponents on "a" go down 1 every term as we go from left
to right. 


The
exponents on "b" go up 1 every term as we go from left to
right. 





This
stuff works real well, but it takes up a lot of space. 


If
I wanted to write out a Pascal's triangle 


for
exponents up to, say, 50, 


It
would take up an amazing amount of space. 


Math
types try to save space anywhere they can. 





They
came up with a shorthand way to tell someone 


what
any level of the triangle would look like. 





It
goes like this ... 





Using
"a" "b" and "n" to stand for 


any
coefficient or exponent we might have ... 





(aX
+ b)^{n} = a^{n}X^{n}  na^{n1}bX^{n1}
... nab^{n1}X + b^{n} 





This
is the start of something called the binomial theorem. 





copyright 2005 Bruce Kirkpatrick 
