Binomial expansion provides a method for expressing powers of binomials like (a + b)n as a sum involving terms with specific coefficients and powers. This concept is essential in algebra and is closely linked to Pascal’s Triangle, which offers a simple way to find the coefficients in each term of the expansion.
According to the binomial theorem, the expansion of a binomial raised to a power follows this general pattern:
(a + b)n can be written as a sum of terms like C(n, r) · an−r · br, where r ranges from 0 to n.
Here, C(n, r) represents the binomial coefficient, also written as \({}^{n}C_{r}
\), and can be found using Pascal’s Triangle or the formula:
C(n, r) = n! / [r!(n - r)!]
Pascal’s Triangle is a number pattern arranged in a triangle, where each entry is formed by adding the two numbers directly above it. Each row corresponds to the coefficients used in binomial expansions.
Below are the initial rows of Pascal’s Triangle, showing the pattern of coefficients used in binomial expansions:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
This expansion follows the formula: (a + b)2 = a2 + 2ab + b2, which is a standard identity for squaring a binomial.
Let a = 2x and b = 3y:
= (2x)2 + 2(2x)(3y) + (3y)2
= 4x2 + 12xy + 9y2
Use the identity: (a - b)2 = a2 - 2ab + b2
Let a = x and b = 5y:
= x2 - 2(x)(5y) + (5y)2
= x2 - 10xy + 25y2
Binomial identities and expansions are important tools in algebra. Using formulas and Pascal’s Triangle, we can quickly expand binomial expressions without manually multiplying. This knowledge is especially useful in algebra, probability, and calculus.
Facebook
Twitter
Linkedin