Binomial Theorem Simple formulas which are helpful to solve algebra problems
What is binomial theorem? The binomial theorem describes the algebraic expansion
of powers of a binomial. i.e., (x + y)n The sum of exponents is equal to n The coefficient a of each term is
a specific positive integer, and are described by Pascal's triangle as shown in the figure.
Binomial Theorem’s Statement According to the theorem
( x + y ) n = nc
0
x n y 0 + nc x n −1 y1 + nc x n − 2 y 2 + ... + nc x1 y n −1 + nc x 0 y n 1
It can be written as,
( x + y)
n −1
2
n
n
n
n
= ∑ nc x n − k y k = ∑ nc x k y n − k k=0
k
k=0
k
By substituting 1 for y, we can get single variable binomial formula
( 1 + x ) n = nc
0
It can be written as,
x 0 + nc x1 + nc x 2 + ... + nc x n −1 + nc x n 1
(1 + x )
n −1
2
n
n
= ∑ nc x k k=0
k
n
Example By substituting n=2 and 4 we will get following formulas respectively.
( x + y)
( x + y)
4
2
= x + 2 xy + y 2
2
= x + 4 x y + 6 x y + 4 xy + y 4
3
2 2
Simple and interesting formulas which are helpful with understanding linear algebra concepts easily.
3
4