CHAPTER XXXVII.
BINOMIAL THEOREM.
406. It may be shown by actual multiplication that
(a + b)(a+c)(a+ d)(a+ e) = a^4 + (b+c+d+e)a^3 + (bc + bd + be + cd + ce + de)a^2 + (bcd + bce + bde + cde)a + bcde (1)
We may, however, write this result by inspection; for the complete product consists of the sum of a number of partial products each of which is formed by multiplying together four letters, one being taken from each of the four factors. If we examine the way in which the various partial products are formed, we see that
(1) The term a^4 is formed by taking the letter a out of each of the factors.
(2) The terms involving a^3 are formed by taking the letter a out of any three factors, in every way possible, and one of the letters b, c, d, e, out of the remaining factor.
(3) The terms involving a^2 are formed by taking the letter a out of any two factors, in every way possible, and two of the letters b, c, d, e, out of the remaining factors.
(4) The terms involving a are formed by taking the letter a out of any one factor, and three of the letters b, c, d, e, out of the remaining factors.
(5) The term independent of a is the product of all the letters b, c, d, e.
Ex. Find the value of (a - 2)(a + 3)(a - 5)(a + 9).
The product =a^4 + (-2+3-5+9)a^3+(-6 + 10 - 18 - 15 + 27 - 45) a^2 + (30 - 54 + 90 - 135)a + 270 = a^4 + 5a^3 - 47a^2 - 69a + 270. 338
