Cambridge Additional Mathematics

270 Counting and the binomial expansion (Chapter 10)

Consider the cube alongside, which has sides of length

(a+b)cm.

The cube has been subdivided into 8 blocks by making 3 cuts

parallel to the cube’s surfaces as shown.

We know that the total volume of the cube is (a+b)^3 cm^3.

However, we can also find an expression for the cube’s volume

by adding the volumes of the 8 individual blocks.

We have: 1 block a£a£a

3 blocks a£a£b

3 blocks a£b£b

1 block b£b£b

) the cube’s volume=a^3 +3a^2 b+3ab^2 +b^3

) (a+b)^3 =a^3 +3a^2 b+3ab^2 +b^3

The sum a+b is called abinomialas it contains two terms.

Any expression of the form (a+b)n is called apower of a binomial.

All binomials raised to a power can be expanded using the same general principles. In this chapter, therefore,

we consider the expansion of the general expression (a+b)n where n 2 N.

Consider the following algebraic expansions:

(a+b)^1 =a+b

(a+b)^2 =a^2 +2ab+b^2

(a+b)^3 =(a+b)(a+b)^2

=(a+b)(a^2 +2ab+b^2 )

=a^3 +2a^2 b+ab^2 +a^2 b+2ab^2 +b^3

=a^3 +3a^2 b+3ab^2 +b^3

Thebinomial expansionof (a+b)^2 is a^2 +2ab+b^2.

Thebinomial expansionof (a+b)^3 is a^3 +3a^2 b+3ab^2 +b^3.

Discovery 2 The binomial expansion

What to do:

1 Expand (a+b)^4 in the same way as for (a+b)^3 above.

Hence expand (a+b)^5 and (a+b)^6.

2 The cubic expansion (a+b)^3 =a^3 +3a^2 b+3ab^2 +b^3 contains 4 terms. Observe that their

coefficients are: 1331

a What happens to the powers ofaandbin each term of the expansion of (a+b)^3?

b Does the pattern inacontinue for the expansions of and (a+b)^6?

F Binomial expansions

ANIMATION

bcm

bcm bcm

acm

acm

acm

(a+b)^4 , (a+b)^5 ,

cyan magenta yellow black

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_10\270CamAdd_10.cdr Monday, 23 December 2013 4:32:44 PM BRIAN