9.4 The Binomial Theorem 401
9.4 The Binomial Theorem There is.a very important and useful formula that involves the natural
numbers and the binomial (a + x). This formula is called the
binomial theorem. The formula gives a short cut for finding values of products such as (a + x)2, (a + x)3, (a + x)4, (a + x)5, and so on.
You may already know the following products. (a + x)2 = (a + x)(a + x) = a2 + 2ax + x2 (a+x)3=(a+x)(a+x)(a÷x)=a3+ 3a2x+ 3ax2+x3 The binomial theorem is stated as follows.
T
H E0RE M
(a + x)
=
where n Note
C(n,O)ax° + C(n,l)a"'x' + C(n,2)a'2x2 + C(n,3)a'3x3 + ... + cn,raxr + ... + C(n,n—1)a'x' + C(n,n)a°f, rN
1 ThevalueofC(n,r)is 2
fl
(n —
,wheren!=n(n—1)(n—2)...(3)(2)(1).
r)! r! The expansion of the product has n + 1 terms.
Example 1 Expand the product (a + x)4.
Solution Use the binomial theorem = C(n,O)ax° + C(n,l)a''x' + C(n,2)a'2x2 + C(n,3)a'3x3 + ... + C(n,r)ax' + ... + C(n,n—l)a'x'' + C(n,n)a°f.
(a +
Here n = 4. Thus,
+ C(4,3)a43x3 + C(4,4)a44x4 x)4 = C(4,O)a4x° + C(4,1)a4'x' + C(4,2)a42x2 4! = Now C(4,O) = recall hat 0.I (4 — O)!O! 4! (a +
C(4,1)=
(4—1)!!! 4!
C(4,2)=
C(4,4) =
_4x3x2x16
(4—2)!2! 2xlx2xl 4!
C(4,3)=
4x3x2x1_4 3x2xlxl
(4 — 3)!3! 4! (4 — 4)!4!
= =1
Therefore, (a + x)4 = a4x° + 4a3x' + 6a2x2 + 4a'x3 + la°x4
=a4+4a3x+6a2x2-i-4ax3+x4 •