# uc3rmw_0905ss

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Hi there.
This is a section summary
covering the binomial theorem.
We can recall that a polynomial that has
exactly two terms is called a binomial.
We first consider these binomial expansions.
We're already familiar with the formulas
for expanding the binomial powers x plus y
squared and x plus y cubed.
In the current section, we can study
a method for expanding x plus y
to the n-th power for
any positive integer n.
We can see from these
first 5 binomial expansions
that as we increase in values of n,
the exponent on the binomial expansion,
that the number of terms increases greatly.
So now, let's look at some properties
for expanding x plus y to the n-th power.
The patterns for expansions of x plus y
to the n-th power where n is equal to 1,
2, 3, 4, and 5, et cetera,
suggests the following,
that the expansion of x plus y to
the n-th power has n plus 1 terms.
We also see that the sum of the exponents
on x and y in each term equals n.
The exponent on x starts at
n, or x to the n-th power
is equal to a to x to the n-th
power times y to the 0-th power
in the first term and
decreases by 1 for each term
until it is 0 in the last term,
where x to the 0 times y to the n
is equal to y to the n-th power.
The exponent on y also starts
at 0, where x to the n-th power
is equal to a to x to the n-th
power times y to the 0-th power
in the first term and increases by 1 for
each term until it is n in the last term.
That is where it's x to the 0 power
times y to the n-th power equals y
to the n-th power.
The variables x and y also have symmetrical
roles, that is replacing x with y and y
with x in the expansion of x
plus y to the n-th power yields
the same terms just in reversed order.
We also can notice that the coefficients
of the first and last terms are both 1.
And the coefficients of the second
and next to last terms are equal.
In general, the coefficients of x to
the n minus j times y to the j and x
to the j times y to the n minus j are
equal for j equals 0, 1, 2, and so on,
all the way to n.
The coefficients in the expansion
of x plus y to the n-th power
are called the binomial coefficients.
Now, let's look at the coefficients
in these binomial expansions
for a review of Pascal's Triangle.
With x plus y to the 0-th
power we have the answer 1.
With x plus y to the first
power we have 1x plus 1y.
With x plus y to the second power, we
get 1x squared plus 2xy plus 1y squared.
And so on.
We can evaluate x plus y to the third
power, x plus y to the fourth power and x
plus y to the fifth power and
we get the following expansions.
We can see that the coefficients in
these expansions are shown in red text.
If we just focus on these coefficients
and leave out the variables we have what's
called Pascal's Triangle.
In Pascal's Triangle we just keep
the coefficients in those expansions
and so we have ones all along the
side here and on the side here.
If we add 1 and 1 here, we get 2
that we put right in this position.
If we add 1 and 2 here we
get 3, which we put here.
If we add 2 and 1 here we get 3 here.
If we add 3 and 3 here we get 6 here.
And so on.
So we can, from the triangle, determine
all the values of the coefficients
for higher and higher values of n.
We note the symmetry in Pascal's Triangle.
If the triangle were folded
vertically down the middle,
the numbers on each side
of the crease would match.
To create a new bottom row in
the triangle put the number 1
in the first and last places of the
new row and add 2 neighboring entries
in the previous row.
The top row is called
the 0-th row, because it
corresponds to the binomial expansion
of x plus y to the 0-th power.
The next row is called
the 1st row, because it
corresponds to the binomial expansion
of x plus y to the 1st power.
All of the rows are names so
that the nth throw corresponds
to the coefficients of x
plus y to the nth power.
The coefficients in a binomial
expansion can be computed
by using ratios of certain factorials.
We first introduced the following symbol.
If r and n are integers with r greater than
or equal to 0, but less than or equal to n,
then we define the symbol n choose
r to be equal to n factorial divided
by r factorial times n minus r factorial.
Or n choose 0 is equal to 1 and
n choose n is also equal to 1.
This symbol is read as n choose r.
And it can be shown that n
choose r is the number of ways
of choosing a subset containing exactly
r elements from a set with n elements.
The numbers n choose r
show up as the coefficients
in the expansion of a binomial power.
For example, x plus y
to the fourth power can
be written as either x to the fourth power
plus 4x cubed times y plus 6x squared y
squared plus 4xy cubed
plus y to the fourth power,
or 4 choose 0 times x to
the fourth power plus 4
choose 1 times x cubed times y
plus 4 choose 2 times s squared
y squared plus 4 choose 3 times
xy cubed and plus 4 choose
4 times y to the fourth power.
This result is the binomial
theorem for n equals
4, which can be proved by
mathematical induction.
So the formal definition of the binomial
theorem is if n is a natural number,
then the binomial expansion
of x plus y to the n-th power
is given by n choose 0 times x to the
n plus n choose 1 times x to the n
minus 1 times y plus n choose 2 times
x to the n minus 2 times y squared.
And so on up to n choose r times x to
the n minus r times y to the r plus
and so on down to n choose
n times y to the nth power,
which we can simplify in the following form.
That is the sum from r equals 0 to
n of n choose r times x to the n
minus r power times y to the r-th power.
The coefficient of x to the
n minus r times y to the r
is n choose r, which is equal to n
factorial divided by r factorial times
n minus r factorial.
The binomial theorem provides an efficient
method for expanding a binomial power.
The binomial theorem can be used
to expand a binomial power directly
without reference to Pascal's Triangle.
This technique is particularly useful
in expanding large powers of a binomial.
For example, the expansion
of x plus y to the 20th power
would require you to produce 20
rows of the Pascal's Triangle.
The binomial coefficients n choose r are
useful in finding a particular coefficient
or term in a binomial expansion.
We have that a term containing the factor
x to the r-th power in the expansion of x
plus y to the nth power is n choose n
minus r times x to the r-th power times y
to the n minus r power.
This term also contains the factor
y to the n minus r-th power.
And so this concludes our section
summary on the binomial theorem.