Binomial Expansion Explained
Learn how to expand expressions of the form (a + b)n using the binomial theorem
What is Binomial Expansion?
The binomial expansion is a way to raise a binomial (an expression with two terms, like a + b) to any power n without having to multiply it out the long way.
Where \(\binom{n}{k}\) is the binomial coefficient, read as "n choose k", and represents the number of ways to choose k elements from a set of n elements.
Simple Example
For example, expanding \((a + b)^3\):
$$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$$
Binomial Coefficients and Pascal's Triangle
Binomial coefficients can be found using Pascal's Triangle, where each number is the sum of the two numbers directly above it.
The nth row of Pascal's Triangle gives the coefficients for expanding \((a + b)^n\).
Try It Yourself
Enter values for a, b, and n to see the binomial expansion:
Key Properties of Binomial Expansion
Number of Terms
The expansion of \((a + b)^n\) has exactly \(n + 1\) terms.
Symmetry
The coefficients are symmetric: \(\binom{n}{k} = \binom{n}{n-k}\)
Sum of Coefficients
The sum of all coefficients is \(2^n\)
Decreasing and Increasing Powers
Powers of a decrease from n to 0, while powers of b increase from 0 to n
Example with Numbers
Let's expand \((2x + 3)^3\):
$$(2x + 3)^3 = (2x)^3 + 3(2x)^2(3) + 3(2x)(3)^2 + (3)^3$$
$$= 8x^3 + 3(4x^2)(3) + 3(2x)(9) + 27$$
$$= 8x^3 + 36x^2 + 54x + 27$$
No comments:
Post a Comment