Binomial
Expression
Any algebraic expression consisting of only two terms is
known as a binomial expression. It's expansion in power of x is known as the
binomial expansion.
e.g. (i) a + x (ii) a2 +
(iii) 4x – 6y
Binomial Theorem
Such formula by which any power of a binomial expression
can be expanded in the form of a series is known as binomial theorem. For a
positive integer n , the expansion is given by .
(a+x)n
= nC0an+nC1an–1
x+nC2 an-2 x2 + . . . + nCr
an–r xr + . . . + nCnxn.
where nC0 , nC1
, nC2 , . . . , nCn are called
Binomial co-efficients. The value of nCr is defined as ![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image002.gif)
Similarly (a – x)n = nC0an–nC1an–1
x+nC2 an-2 x2 – . . . +(–1)r
nCr an–r xr+ . . . +(–1)n
nCnxn.
Example.1
Expand![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image003.gif)
Solution
= 7C0x7+7C1x6
+7C2x5
+7C3x4
+ 7C4x3
![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image008.gif)
+7C5x2
+7C6
x
+7C7![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image011.gif)
= x7 + 7x5
+ 21x3 + 35 x +
+
+
+
.
General Term in the
Expansion:
The general term in
the expansion of ( a+x)n is (r+1)th term given as tr+1
= nCr an-r xr Similarly the
general term in the expansion of ( x + a)n given as tr+1
= nCr xn-r ar. The terms are
considered from the beginning. .
The (r + 1)th
term from the end = ( n – r + 1)th term from the beginning . .
Corollary:
Coefficient of xr in expansion
of (1 + x)n is nCr .
Example.1
Find the
co-efficient of x24 in
.
Solution
Since, general term
((r+1) th term) in
= 15Cr(x2)15–r
![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image017.gif)
= 15Cr
x30–2r
= 15Cr
3rar x30–3r
If this term
contains x24. Then .
30–3r = 24
&⇒ 3r = 6
&⇒ r = 2
Therefore, the
co-efficient of x24 = 15C2 ´9a2.
Example.2
If the co-efficient
of (2r+4)th term and (r–2) th term in the expansion of (1+x)18 are
equal, find r .
Solution
Since, co-efficient
of (2r+4) th term in (1+x)18 = 18C2r+3
Co-efficient of
(r–2) th term = 18Cr–3
&⇒ 18C2r+3 = 18Cr–3
&⇒ 2r+3+r–3 = 18
&⇒ 3r = 18
&⇒ r = 6
Coefficients of Equidistant
Terms from Begnning and end:
The binomial
coefficients in the expansion of (a+x)n equidistant from the
beginning and the end are equal. .
Middle Terms:
·
When
n is even
Middle term of the expansion is
term.
·
When
n is odd
In this case
term and
term are the middle
terms.
e.g. Middle term in the expansion of (1
+ x)4 and (1 + x)5.
Expansion of (1
+ x)4 has 5 terms, so third term is middle term i.e.
term.
Expansion of (1 + x)5 has 6
terms, so 3rd and 4th both are middle terms, that is,
th and
th terms are middle
terms.
Exapmle.1
Find the middle
term in the expression of ( 1- 2x + x2)n.
Solution
( 1- 2x + x2)n
= [ ( 1- x)2]n = ( 1- x)2n
Here 2n is even
integer, therefore,
th i.e. (
n+1) th term will be the middle term
Now (n+1) th term
in (1 - x)2n = 2nCn (1)2n-n(-x)n
= 2nCn(-x)n
=
.
Example.2
Prove that middle
term in the expansion of
is
.
Solution
Since 2n is even,
therefore
th i.e. (n+1)th term
will be the middle term.
Now (n+1) th term
i.e. middle term in
is given
by
tn+1 = 2nCnx2n-n
=2nCn
xn![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image031.gif)
= ![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image032.gif)
= ![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image033.gif)
=
.
Greatest Binomial
Coefficient:
To determine the greatest coefficient in the
binomial expansion of (1 + x)n , consider the following
= ![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image036.gif)
Now
the (r + 1)th binomial coefficient will be greater than the rth
binomial coefficient when,
=
>1
&⇒
.......(1)
But r must be an integer, and therefore when
n is even, the greatest binomial coefficient is given by the greatest value of
r , consistent with (1) i.e., r =
and hence the greatest
binomial coefficient is nCn/2.
Similarly if n be odd, the greatest binomial
coefficient is given when,
![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image039.gif)
Example.3
Show that the
greatest coefficient in the expansion of
is
.
Solution
Since middle term
has the greatest coefficient,
So, greatest
coefficient = coefficient of middle term
= 2nCn
=
.
Greatest Term:
To determine the
numerically greatest term (absolute value) in the expansion of (a + x)n,
when n is a positive integer. Consider.
![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image042.gif)
Thus |Tr + 1|
> |Tr| if ![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image043.gif)
.
. . (2)
Note:
.ThusTr+1
will be the greatest term if, r has the greatest value consistent with
inequality (2)
Example.1
Find the greatest
term in the expansion of (2 + 3x)9 if x = 3/2.
Solution
![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image047.gif)
![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image048.gif)
Therefore Tr+1
≥ Tr if,
90 - 9r ≥ 4r
&⇒ 90 ≥
13r
, r being an integer, hence r = 6.
Tr+1 = T7
= T6+1 = 9C6 (2)3 (3x)6 ![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image050.gif)
Example.2
Find the greatest
term in the expansion of
.
Solution
Let rth term be
the greatest term.
Since
, now ![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image053.gif)
&⇒
.
. . . (1)
Now again
£ 1
&⇒
. . . (2)
From (1) and (2) follows that
![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image057.gif)
&⇒ r = 8 is the greatest term and
its value
=
=
.
Properties of Binomial
Expansion:
There are (n + 1)
terms in the expansion of (a+b)n, the first and the last term being
an and bn respectively.
Example.1 .
If in the expansion of ( 1+x)43,
the coefficient of (2r+1)th term is equal to the coefficient of (r+2)th term
find r.
Solution
Given in the
expansion of (1+ x)43, the coefficient of (2r +1)th term = the
coefficient of (r +1)th term so, 43C2r = 43Cr+1
&⇒ 2r + r + 1 = 43 or r = 14.
for the binomial expansion (a + x)n.
The Pascal’s
triangle is way of directly calculating binomial coefficients for different
indices. It is given as follows.
![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image061.jpg)
How to construct a
Pascal's triangle is explained below
![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image062.jpg)
The coefficients in
the Binomial Expansion can be easily determined with the help of Pascal's
triangle.
- Dm
(ax + b)n = 0 if m > n where Dm is mth
derivative w. r. t. x . .
- Dm (ax + b)n = ann!
if n = m
Dm (ax + b)n ![](http://www.quizsolver.com/radix/dth/notif/Binomial%20Expression_1_files/image063.gif)