TERMS USED IN MATH

COEFFICIENTS

• The coefficients of the binomial expansion are called binomial coefficients. The coefficients have symmetry. The coefficient of xn–ryr in the expansion of (x + y)n is written or nCr . (x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5 The first and last coefficients are 1. The coefficients of the second and second to last terms are equal to n. 1 1 Example: What are the last 2 terms of (x + y)10 ? Since n = 10, the last two terms are 10xy9 + 1y10.

THE BINOMIAL THEROM

• The binomial theorem provides a useful method for raising any binomial to a nonnegative integral power. Consider the patterns formed by expanding (x + y)n. (x + y)0 = 1 (x + y)1 = x + y (x + y)2 = x2 + 2xy + y2 (x + y)3 = x3 + 3x2y + 3xy2 + y3 (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 (x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5 Notice that each expansion has n + 1 terms. 1 term 2 terms 3 terms 4 terms 5 terms 6 terms Example: (x + y)10 will have 10 + 1, or 11 terms.

PASCHALS TRIANGLE

• The triangular arrangement of numbers below is called Pascal’s Triangle. Each number in the interior of the triangle is the sum of the two numbers immediately above it. The numbers in the nth row of Pascal’s Triangle are the binomial coefficients for (x + y)n .