1076 CHAPTER 11 Further Topics in Algebra Refer again to Pascal’s triangle. Notice the symmetry in each row. This suggests that binomial coefficients should have the same property. That is, an r b = a n n −rb . This is true because the numerators are equal and the denominators are equal by the commutative property of multiplication. an r b = n! r!1n −r2! and a n n −rb = n! 1 n −r2! r! Binomial Theorem For any positive integer n and any complex numbers x and y, 1x + y2n is expanded as follows. 1x +y2n =xn + a n 1b xn−1 y + a n 2b xn−2 y2 + a n 3b xn−3 y3 + P + a n rb xn−r yr +P+ a n n −1b xyn−1 +yn The Binomial Theorem Our observations about the expansion of 1x + y2n are summarized as follows. 1. There are n + 1 terms in the expansion. 2. The first term is xn, and the last term is yn. 3. In each succeeding term, the exponent on x decreases by 1 and the exponent on y increases by 1. 4. The sum of the exponents on x and y in any term is n. 5. The coefficient of the term with xryn-r or xn-ryr is An r B. These observations suggest the binomial theorem. LOOKING AHEAD TO CALCULUS The binomial theorem is used to show that the derivative of ƒ1x2 = x n is given by the function ƒ′1x2 = nx n-1. This fact is used extensively in calculus. NOTE The binomial theorem may also be written as a series using summation notation. 1x +y2n = a n r=0 a n r b xn−ryr In agreement with Pascal’s triangle, the coefficients of the first and last terms are both 1. That is, an 0b = a n nb =1.
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