Intermediate Algebra (11th edition)

SECTION 12.4 The Binomial Theorem 701

OBJECTIVES

The Binomial Theorem

12.4

1 Expand a binomial

raised to a power.

2 Find any specified

term of the

expansion

of a binomial.

OBJECTIVE 1 Expand a binomial raised to a power. Observe the expansion

of the expression for the first six nonnegative integer values of n.

Expansions of

By identifying patterns, we can write a general expansion for

First, if nis a positive integer, each expansion after begins with xraised

to the same power to which the binomial is raised. That is, the expansion of

has a first term of the expansion of has a first term of and so on.

Also, the last term in each expansion is yto this same power, so the expansion of

should begin with the term and end with the term

The exponents on xdecrease by 1 in each term after the first, while the exponents

on y, beginning with yin the second term, increase by 1 in each succeeding term.

Thus, the variablesin the expansion of have the following pattern.

This pattern suggests that the sum of the exponents on xand yin each term is n. For

example, in the third term shown, the variable part is and the sum of the expo-

nents, and 2, is n.

Now examine the pattern for the coefficientsof the terms of the preceding expan-

sions. Writing the coefficients alone in a triangular pattern gives Pascal’s triangle,

named in honor of the 17th-century mathematician Blaise Pascal.

n- 2

xn-^2 y^2

xn, xn-^1 y, xn-^2 y^2 , xn-^3 y^3 , Á, xyn-^1 , yn

1 x+ y 2 n

1 x+y 2 n xn yn.

x^1 , 1 x+y 22 x^2 ,

1 x+ y 21

1 x+y 20

1 x+ y 2 n.

1 x+y 25 = x^5 + 5 x^4 y+ 10 x^3 y^2 + 10 x^2 y^3 + 5 xy^4 + y^5

1 x+y 24 = x^4 + 4 x^3 y+ 6 x^2 y^2 + 4 xy^3 +y^4 ,

1 x+y 23 = x^3 + 3 x^2 y+ 3 xy^2 + y^3 ,

1 x+y 22 = x^2 + 2 xy+y^2 ,

1 x+y 21 = x+y, 1 x+y 2 n

1 x+y 20 = 1,

1 x+y 2 n

Blaise Pascal (1623–1662)

Pascal’s Triangle

1

11

121

1331

14641

1 5 10 10 5 1 and so on

The first and last terms of each row are 1. Each number in the interior of the triangle is the

sum of the two numbers just above it (one to the right and one to the left). For example, in

the fifth row from the top, 4 is the sum of 1 and 3, 6 is the sum of 3 and 3, and so on.

To obtain the coefficients for we attach the seventh row to the table by

starting and ending with 1, and adding pairs of numbers from the sixth row.

1615201561 Seventh row

We then use these coefficients to expand as

1 x+y 26 = x^6 + 6 x^5 y+ 15 x^4 y^2 + 20 x^3 y^3 + 15 x^2 y^4 + 6 xy^5 + y^6.

1 x+ y 26

1 x+ y 26 ,