Cambridge Additional Mathematics

Counting and the binomial expansion (Chapter 10) 271

c Write down the triangle of

coefficients to row 6 :

n=1

n=2

n=3

11

121

1331

..

.

row 3

3 The triangle of coefficients incabove is calledPascal’s triangle. Investigate:

a the predictability of each row from the previous one

b a formula for finding the sum of the numbers in thenth row of Pascal’s triangle.

4aUse your results from 3 to predict the elements of the 7 th row of Pascal’s triangle.

b Hence write down the binomial expansion of (a+b)^7.

c Check your result algebraically by using (a+b)^7 =(a+b)(a+b)^6 and your results from 1.

From theDiscoverywe obtained (a+b)^4 =a^4 +4a^3 b+6a^2 b^2 +4ab^3 +b^4

=a^4 +4a^3 b^1 +6a^2 b^2 +4a^1 b^3 +b^4

Notice that: ² As we look from left to right across the expansion, the powers ofadecrease by 1 ,

while the powers ofbincrease by 1.

² The sum of the powers ofaandbin each term of the expansion is 4.

² The number of terms in the expansion is 4+1=5.

² The coefficients of the terms are row 4 of Pascal’s triangle.

For the expansion of (a+b)n where n 2 N:

² As we look from left to right across the expansion, the powers ofadecreaseby 1 , while the powers

ofbincreaseby 1.

² The sum of the powers ofaandbin each term of the expansion isn.

² The number of terms in the expansion is n+1.

² The coefficients of the terms are rownof Pascal’s triangle.

In the following examples we see how the general binomial expansion (a+b)n may be put to use.

Example 15 Self Tutor

Using (a+b)^3 =a^3 +3a^2 b+3ab^2 +b^3 , find the binomial expansion of:

a (2x+3)^3 b (x¡5)^3

a In the expansion of (a+b)^3 we substitute a=(2x)

and b= (3).

) (2x+3)^3 =(2x)^3 + 3(2x)^2 (3) + 3(2x)^1 (3)^2 + (3)^3

=8x^3 +36x^2 +54x+27

b We substitute a=(x) and b=(¡5)

) (x¡5)^3 =(x)^3 +3(x)^2 (¡5) + 3(x)(¡5)^2 +(¡5)^3

=x^3 ¡ 15 x^2 +75x¡ 125

Brackets are essential!

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_10\271CamAdd_10.cdr Monday, 23 December 2013 4:32:53 PM BRIAN