The document discusses expanding binomial expressions like (x + y)n using Pascal's triangle and the binomial theorem. It explains that each term in the expansion has exponents of x and y that add up to n, with the x exponent decreasing by 1 and the y exponent increasing by 1 in subsequent terms. The coefficients of the terms form Pascal's triangle. It also presents the binomial theorem formula for finding the coefficients and discusses using factorials. It provides an example of finding a specific term in a binomial expansion by identifying which value of k corresponds to that term number.
2. Expanding Binomials
- in this section we will look at ways to expand
binomial expressions like:
(x + y)5
(2x – 3y)7
We will do this WITHOUT having to multiply the
expressions out
3. pascal's TrianglE
Consider (x + y)n
(x + y)0
= 1 which we can think of as 1x0
y0
(x + y)1
= x + y OR 1x1
y0
+ 1x0
y1
(x + y)2
= x2
+ 2xy + y2
= 1x2
y0
+ 2x1
y1
+ 1x0
y2
(x + y)3
= x3
+ 3x2
y + 3xy2
+ y3
OR 1x3
y0
+ 3x2
y1
+ 3x1
y2
+ 1x0
y3
Notice that for each term in the expansion…
The exponents add up to “n”
Also, for each subsequent term, the exponent for x decreases
by 1 while the exponent for y INCREASES by one
Additionally, the coefficients for the terms form a pattern
known as Pascals’ Triangle (see board)
4. To pErform a Binomial
Expansion:
Go to the appropriate row in Pascal’s triangle to
obtain the coefficients
Write out terms with the variables, remembering that
the powers add up to n for each term, start with xn
y0
,
end with x0
yn
See the example on the next slide
5. paTTErns in ThE Expansion of
(x + y)n
There are n + 1 terms
The exponent n of (x+y)n
is the exponent of x in
the 1st
term and the exponent of y in the last term
In successive terms, the exponent of x decreases
by 1 and the exponent of y increases by 1
The sum of the exponents in each term is n
The coefficients are symmetric: They increase at
the beginning of the expansion and decrease at
the end
6. Expand
Write row 5 of Pascal’s triangle.
1 5 10 10 5 1
Use the patterns of a binomial expansion and the
coefficients to write the expansion of
Answer:
8. The Binomial Theorem
Another way to show the coefficients in a binomial
expansion
If n is a nonnegative integer, then (a + b)n
= 1an
b0
+
(n/1)an-1
b1
+ (n(n-1)/(1*2)an-2
b2
+ (n(n-1)(n-2))/(1*2*3) an-3
b3
+ … 1a0
bn
9. The expression will have nine terms. Use the sequence
to find the coefficients
for the first five terms. Use symmetry to find the
remaining coefficients.
Expand
12. Factorials
The factors in the coefficients of a binomial
expansion involve special products called
FACTORIALS
For example, the product 4 * 3 * 2 * 1 is written 4!
and is read “4 factorial”
In general, if n is a positive integer, then n! equals
n * (n – 1) * (n – 2) * (n – 3) *…2*1
(By definition, 0! = 1)
If a rational expression contains some factorials,
often a number of terms will cancel out
14. Binomial Theorem, facTorial form
0 1 1 2 2 0! ! ! !
( ) ...
!0! ( 1)!1! ( 2)!2! 0! !
n n n n nn n n n
x y x y x y x y x y
n n n n
− −
+ = + + + +
− −
0
!
( ) *
( )! !
n
n n k k
k
n
x y x y
n k k
−
=
+ =
−
∑
19. Finding a speciFic term
Sometimes you are only asked to find one term in an
expansion
Note that when the Binomial expansion is written
using Sigma notation, k = 0 for the 1st
term, k = 1 for
the 2nd
term, k = 2 for the 3rd
term, and so on.
In general, the value of k is one less than the number
of the term you are finding!
20. Find the fourth term in the expansion of
First, use the Binomial Theorem to write the expression
in sigma notation.
In the fourth term,
