Hello class, before we move on to calculus, we have a small topic on Binomial Theorem to cover.
BINOMIAL THEOREM
(x + y)4 =
(x + y)2 =
A binomial is a mathematical expression containing two terms. If raised to an integer power, it can be expanded in this manner. Notice how the power of the first variable decreases by 1 when moving from term to term, while that of the second increases.
We can use this array of numbers called the Pascal's triangle, where the row number is the exponent, while the numbers in each row represent the coefficients for the respective terms.
11 2 11 3 3 11 4 6 4 11 5 10 10 5 1
This enables us to use the nCr notation, to find the coefficient of the rth term in a binomial, starting from 0, among n terms, which is is always 1 more than the exponent.
Alternatively, we are able to obtain the coefficients using another method, which involves the concept of factorials, the product of consecutive descending numbers.
nCr =
n
r
( )
=
n!
r! (n-r!)
With these 2 concepts, we can derive the general term of an expanded binomial. This is particularly useful if we just want to identify a particular term in the whole expression.
Tr+1 =
In (x + y)n ,
r
n
( )
xn-r
yr
OK class, now take out your A4 paper, we will do some written work, 20 minutes.