Binomial Theorem I

Learning Objectives State the Binomial Theorem for positive integer exponents. Relate the coefficients in a binomial expansion to Pascal's Triangle. Calculate binomial coefficients using the combination formula, nCr. Expand binomials of the form (a+b)^n using the Binomial Theorem. Determine a specific term in a binomial expansion without full expansion. Find the coefficient of a specific power of a variable in an expansion. How would you expand (x + y)^15 without multiplying it out 15 times? 🤔 There's a powerful shortcut! This lesson introduces the Binomial Theorem, a fundamental formula in algebra for expanding binomials raised to any positive integer power. You will learn how to use combinations and patterns to quickly find the expanded form, a specific term, o...

TermDefinitionExample BinomialA polynomial with exactly two terms.x + 3, 2a - b, 4x^2 + 5y^3 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 CombinationA selection of items from a set where the order of selection does not matter. It is denoted as C(n, r) or nCr, read as 'n choose r'.The number of ways to choose 2 letters from {A, B, C} is 3C2 = 3. The combinations are {A,B}, {A,C}, and {B,C}. Binomial CoefficientThe coefficients of the terms in a binomial expansion. Each coefficient can be calculated using the combination formula nCr.In the expansion of (a+b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3, the binomial coefficients are 1, 3, 3, 1. Pascal's TriangleA triangular...

The Binomial Theorem Formula (a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r = \binom{n}{0}a^n b^0 + \binom{n}{1}a^{n-1}b^1 + ... + \binom{n}{n}a^0 b^n This is the general formula for expanding any binomial (a+b) raised to a positive integer power 'n'. The symbol \binom{n}{r} is the same as nCr. Combination Formula (Binomial Coefficient) \binom{n}{r} = C(n,r) = \frac{n!}{r!(n-r)!} Use this formula to calculate any specific binomial coefficient in the expansion, where 'n' is the exponent of the binomial and 'r' is the exponent of the second term (starting from r=0). General Term Formula T_{r+1} = \binom{n}{r} a^{n-r} b^r This formula is a powerful shortcut used to find any specific term in the expansion without calculating the entire se...