Pascal's triangle and the Binomial Theorem

Learning Objectives Construct the first several rows of Pascal's triangle and identify key patterns. Define and calculate binomial coefficients using both Pascal's triangle and the combination formula. State the Binomial Theorem for positive integer exponents. Apply the Binomial Theorem to expand binomials of the form (a + b)^n. Expand binomials with coefficients and negative terms, such as (2x - 3y)^n. Find a specific term in a binomial expansion without calculating the entire expansion. Ever tried to calculate (x + y)^7 by hand? 🤯 There's a much faster way, and it's hidden inside a simple triangle of numbers. This tutorial will introduce you to two powerful tools: Pascal's triangle and the Binomial Theorem. You will learn how these concepts provi...

TermDefinitionExample BinomialA polynomial expression with exactly two terms.(x + 5), (2a - 3b), or (x^2 + 1) Pascal's TriangleA triangular array of numbers in which the first and last number of each row is 1, and every other number is the sum of the two numbers directly above it.Row 4 is 1, 4, 6, 4, 1. The '6' is the sum of the '3' and '3' from the row above it. FactorialThe product of an integer and all the positive integers below it, denoted by an exclamation mark (!). By definition, 0! = 1.5! = 5 × 4 × 3 × 2 × 1 = 120 Binomial CoefficientThe coefficients of the terms in a binomial expansion. It represents the number of ways to choose k elements from a set of n elements, and is written as C(n, k) or \binom{n}{k}.The coefficient of the x^2y^2 term in t...

Binomial Coefficient Formula (Combinations) \binom{n}{k} = C(n, k) = \frac{n!}{k!(n-k)!} Use this formula to calculate the coefficient of any term in a binomial expansion. 'n' is the exponent of the binomial, and 'k' is the exponent of the second term in the expansion (starting from k=0). The Binomial Theorem (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k This is the general formula for expanding any binomial (a+b) raised to a positive integer power 'n'. It states that the expansion is the sum of terms, where for each term you calculate the binomial coefficient and the appropriate powers of 'a' and 'b'. Formula for the (k+1)th Term T_{k+1} = \binom{n}{k} a^{n-k} b^k A direct application of the Binomial Theorem used...