bionomial theorem
A very good topic.
The binomial theorem is a formula that allows us to expand a binomial raised to a positive integer power. It helps us find the coefficients of each term in the expanded form. The general form of the binomial theorem is:
(x + y)^n = C(n,0) x^n y^0 + C(n,1) x^(n-1) y^1 + C(n,2) x^(n-2) y^2 + ... + C(n,n-1) x^1 y^(n-1) + C(n,n) x^0 y^n
Where:
- (x + y) represents the binomial expression
- n is a positive integer exponent
- C(n,k) represents the binomial coefficient, which is the number of ways to choose k items from a set of n items.
To find the binomial coefficients, you can use Pascal's triangle or the formula:
C(n,k) = n! / (k! * (n - k)!)
Here's an example to illustrate how to use the binomial theorem:
Example:
Expand (x + y)^4.
Using the binomial theorem, we have:
(x + y)^4 = C(4,0) x^4 y^0 + C(4,1) x^3 y^1 + C(4,2) x^2 y^2 + C(4,3) x^1 y^3 + C(4,4) x^0 y^4
= (1) x^4 y^0 + (4) x^3 y^1 + (6) x^2 y^2 + (4) x^1 y^3 + (1) x^0 y^4
= x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4
So, (x + y)^4 can be expanded as x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4.
That's how we use the binomial theorem to expand a binomial expression.