Cambridge International AS and A Level Mathematics Pure Mathematics 1

Binomial expansions

P1^

3

ExamPlE 3.18 The first three terms in the expansion of ()ax+bx

6

where a  0, in descending

powers of x, are 64x^6 – 576x^4 + cx^2. Find the values of a, b and c.

SOlUTION

Find the first three terms in the expansion in terms of a and b:

()ax+bx = ax ax bx









() +









()()+





(^6665) 
0

6

1

6

2 

()()

=+ +

ax bx

ax abxabx

4 2

66 6154 5422

So a^6 x^6 + 6a^5 bx^4 + 15 a^4 b^2 x^2 = 64 x^6 − 576 x^4 + cx^2

Compare the coefficients of x^6 : a^6 = 64 ⇒ a = 2

Compare the coefficients of x^4 : 6 a^5 b = − 576

Since a = 2 then 192b = − 576 ⇒ b = − 3

Compare the coefficients of x^2 : 15 a^4 b^2 = c

Since a = 2 and b = −3 then c = 15 × 24 × (–3)^2 ⇒ c = 2160

●?^ A Pascal puzzle

1.1^2 = 1.21 1.1^3 = 1.331 1.1^4 = 1.4641

What is 1.1^5?

What is the connection between your results and the coefficients in Pascal’s

triangle?

Relationships between binomial coefficients

There are several useful relationships between binomial coefficients.

Symmetry

Because Pascal’s triangle is symmetrical about its middle, it follows that

n

r

n

nr









=

−









.

x^4 ×=x^12 x^2

Remember both

26 = 64 and (–2)^6 = 64,

but as a > 0 then a = 2.