1073 11.4 The Binomial Theorem 11.4 The Binomial Theorem ■ A Binomial Expansion Pattern ■ Pascal’s Triangle ■ n-Factorial ■ Binomial Coefficients ■ The BinomialTheorem ■ k thTerm of a Binomial Expansion Pascal’s Triangle Row 1 0 1 1 1 1 2 1 2 1 3 3 1 3 1 4 6 4 1 4 1 5 10 10 5 1 5 With the coefficients arranged in this way, each number in the triangle is the sum of the two numbers directly above it (one to the right and one to the left). LOOKING AHEAD TO CALCULUS Students taking calculus study the binomial series, which follows from Isaac Newton’s extension to the case where the exponent is no longer a positive integer. His result led to a series for 11 + x2k, where k is a real number and x 61. A Binomial Expansion Pattern In this section, we introduce a method for writing the expansion of expressions of the form 1x + y2n, where n is a natural number. Some expansions for various nonnegative integer values of n follow. 1x + y20 = 1 1x + y21 = x + y 1x + y22 = x2 + 2xy + y2 1x + y23 = x3 + 3x2y + 3xy2 + y3 1x + y24 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 1x + y25 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5 Notice that after the special case 1x + y20 = 1, each expansion begins with x raised to the same power as the binomial itself. That is, the expansion of 1x + y21 has a first term of x1, 1x + y22 has a first term of x2, 1x + y23 has a first term of x3, and so on. Also, the last term in each expansion is y to the same power as the binomial. Thus, the expansion of 1x + y2n should begin with the term x n and end with the term y n. Notice that the exponent on x decreases by 1 in each term after the first, while the exponent on y, beginning with y in the second term, increases by 1 in each succeeding term. That is, the variables in the terms of the expansion of 1x + y2n have the following pattern. x n, x n-1y, x n-2y 2, x n-3y 3, c , xy n-1, y n This pattern suggests that the sum of the exponents on x and y in each term is n. For example, the third term in the list above is x n-2y2, and the sum of the exponents is n - 2 + 2 = n. Pascal’s Triangle Now, examine the coefficients in the terms of the expansion of 1x + y2n. Writing the coefficients alone gives the following pattern.
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