Binomial expansion

1.5 BINOMIAL EXPANSION
x = −1 gives respectively
52
C
D1
2
=
−
+ ,
24
6
2
4
36
B+C
=
− D1 + 2,
7
7
86
C−B
D1
2
=
−
+ .
63
7
3
9
These equations reduce to
4C − 12D1 = 40,
B + C − 7D1 = 22,
−9B + 9C − 21D1 = 72,
with solution B = 0, C = 1, D1 = −3.
Thus, finally, we may rewrite the original expression F(x) in partial fractions as
F(x) = x + 2 +
1
3
2
.
−
+
x2 + 6 x − 2 (x − 2)2
1.5 Binomial expansion
Earlier in this chapter we were led to consider functions containing powers of
the sum or difference of two terms, e.g. (x − α)m . Later in this book we will find
numerous occasions on which we wish to write such a product of repeated factors
as a polynomial in x or, more generally, as a sum of terms each of which contains
powers of x and α separately, as opposed to a power of their sum or difference.
To make the discussion general and the result applicable to a wide variety of
situations, we will consider the general expansion of f(x) = (x + y)n , where x and
y may stand for constants, variables or functions and, for the time being, n is a
positive integer. It may not be obvious what form the general expansion takes
but some idea can be obtained by carrying out the multiplication explicitly for
small values of n. Thus we obtain successively
(x + y)1 = x + y,
(x + y)2 = (x + y)(x + y) = x2 + 2xy + y 2 ,
(x + y)3 = (x + y)(x2 + 2xy + y 2 ) = x3 + 3x2 y + 3xy 2 + y 3 ,
(x + y)4 = (x + y)(x3 + 3x2 y + 3xy 2 + y 3 ) = x4 + 4x3 y + 6x2 y 2 + 4xy 3 + y 4 .
This does not establish a general formula, but the regularity of the terms in
the expansions and the suggestion of a pattern in the coefficients indicate that a
general formula for power n will have n + 1 terms, that the powers of x and y in
every term will add up to n and that the coefficients of the first and last terms
will be unity whilst those of the second and penultimate terms will be n.
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PRELIMINARY ALGEBRA
In fact, the general expression, the binomial expansion for power n, is given by
(x + y)n =
k=n
n
Ck xn−k y k ,
(1.49)
k=0
where n Ck is called the binomial coefficient and is expressed in terms of factorial
functions by n!/[k!(n − k)!]. Clearly, simply to make such a statement does not
constitute proof of its validity, but, as we will see in subsection 1.5.2, (1.49) can
be proved using a method called induction. Before turning to that proof, we
investigate some of the elementary properties of the binomial coefficients.
1.5.1 Binomial coefficients
As stated above, the binomial coefficients are defined by
n
n!
n
≡
for 0 ≤ k ≤ n,
Ck ≡
k
k!(n − k)!
(1.50)
where in the second identity we give a common alternative notation for n Ck .
Obvious properties include
(i) n C0 = n Cn = 1,
(ii) n C1 = n Cn−1 = n,
(iii) n Ck = n Cn−k .
We note that, for any given n, the largest coefficient in the binomial expansion is
the middle one (k = n/2) if n is even; the middle two coefficients (k = 12 (n ± 1))
are equal largest if n is odd. Somewhat less obvious is the result
n
n!
n!
+
k!(n − k)! (k − 1)!(n − k + 1)!
n![(n + 1 − k) + k]
=
k!(n + 1 − k)!
(n + 1)!
= n+1 Ck .
=
k!(n + 1 − k)!
Ck + n Ck−1 =
(1.51)
An equivalent statement, in which k has been redefined as k + 1, is
n
Ck + n Ck+1 = n+1 Ck+1 .
(1.52)
1.5.2 Proof of the binomial expansion
We are now in a position to prove the binomial expansion (1.49). In doing so, we
introduce the reader to a procedure applicable to certain types of problems and
known as the method of induction. The method is discussed much more fully in
subsection 1.7.1.
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