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EDEN UNIVERSITY
SCHOOL OF NATURAL SCIENCES
BINOMIAL EXPANSIONS (MAT1320) TUTORIAL
LECTURER: MR R DOZVA
The binomial theorem states that
(a + b)n^ = n 0
! an^ + n 1
! an−^1 b + n 2
! an−^2 b^2 + · · · + n r
! an−rbr^ + · · · + n n
! bn^ (1)
or briefly (a + b)n^ =
X^ n r=
n r
! an−rbr. (2)
Recall that the binomial coefficients are evaluated as n r
! = n! r!(n − r)!
General Term:
The term n r
! an−rbr^ in the expansion of the binomial theorem is called the general term or
the (r + 1)th term. It is denoted by Tr+1. Hence it is given by
Tr+1 = nr
! an−rbr
Note: The General term is used to find out the specified term or the required coefficient of the term in the binomial expansion.
Example: Find the eighth term in the expansion of
2 x^2 − (^) x^12
12 .
Solution:
2 x^2 − (^) x^12
12 .
The general term is, Tr+1 = n r
! an−rbr.
Here, T 8 =? a = 2x^2 , b = − 1 x^2
, n = 12, r = 7,
Therefore, T7+1 = 12 7
! (2x^2 )^12 −^7
− 1 x^2
7 .
After simplifying, we get the eighth term as
T 8 = −
x^4.
Tutorial Set:
- Express 9 × 8 × 7 × 6 4 × 3 × 2 × 1 in factorial notation.
- Factorize 8! − 5(7!)
- Factorize (n + 1)! + n^2 (n − 1)!
- Write the following in factorial notation
(a) n(n^ −^ 1)(n^ −^ 2) 3 × 2 × 1 (b) (n^ −^ 1)(n^ −^ 2)(n^ −^ 3)(n^ −^ 4) 4 × 3 × 2 × 1
- Using the binomial theorem, show that
3 x − (^1) x
4 = 81x^4 − 108 x^2 + 54 − (^12) x 2 + x^14
- Expand the binomial
x +^2 x
4 by the binomial theorem.
Middle Term in the Expansion of (a + b)n
In the expansion of (a + b)n, there are (n + 1) terms.
CASE 1:
If n is even then (n + 1) will be odd, so
(^) n 2 + 1
th term will be the only one middle term in the expansion. For example, if n = 8 (even), the number of terms will be 9 (odd), therefore
= 5th term.
CASE 2:
If n is odd then (n + 1) will be even, in this case there will not be a single term, but n + 1 2
th and
(^) n + 1 2 + 1
th term will be the two middle terms in the expansion.
For example, if n = 9 (odd), the number of terms will be 10, i.e
th and
i.e 5th^ and 6th^ term are found by using the formula for the general term.
