Back to: Further Mathematics SS2
PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITVE AND FRACTIONAL POWER
PASCAL’S TRIANGLE
Consider the expressions of each of the following:
(x + y)0; (x + y )1; (x + y)2; (x + y)3; (x + y)4
(x + y)0 = 1
(x + y)1 = 1x + 1y
(x + y)2 = 1x2 + 2xy + 1y2
(x + y)3= 1x3 + 3x2y + 3xy2 + 1y3
(x + y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1x4
The coefficient of x and y can be displayed in an array as:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
The array of coefficients displayed above is called Pascal’s triangle, and it is used in determining the co-efficients of the terms of the powers of a binomial expression
Coefficient of (x + y)0 1
Coefficient of (x + y)1 1 1
Coefficients of (x + y)2 1 2 1
Coefficients of (x + y)3 1 3 3 1
Coefficients of (x + y)4 1 4 6 4 1
Example 1
Using Pascal’s riangle, expand and simplify completely: (2x + 3y)4
Solution:
(2x + 3y)4 = (2x)4 + 4(2x)3 (3y) + 6(2x)2(3y)2 + 4(2x)(3y)3 + (3y)4
= 16x4 + 96x3y + 216x2y2 + 216xy3 + 81y4
Examples 2:
Using pascal’s triangle, the coefficients of (x + y)5are: 1,5,10,10,5,1.
Therefore (x – 2y)5 = x5 + 5x4(-2y) + 10x3(-2y)2 + 10x2(-2y)3 + 5x(-2y)4 + (-2y)5
= x5 – 10x4y + 40x3y2 – 80x2y3 + 80xy4 – 32y5
Example 3
Using Pascal’s triangle, simplify, correct to 5 decimal places (1.01)4
Solution
We can write (1.01)4 = (1 + 0.01)4
(1 + 0.01)4 = 1 + 4(0.01) + 6(0.01)2 + 4(0.01)3+(0.01)4
= 1 + 0.04 + 0.0006 + 0.000004 + 0.00000001
= 1.04060401
= 1.04060 (5 d.p)
It can be shown that the binomial expansion formula holds for positive, negative, integral or any rational value of n, provided there is a restriction on the values of x and y in the expansion of (x + y)n
We shall however consider only the binomial expansion formula for a positive integral n
Example 4:
a. Write down the binomial expansion of 6 simplifying all the terms
b. Use the expansion in (a) to evaluate (1.0025)6 correct to five significant figures.
(1.03)6 + (0.97)6 = 2 + 270(0.01)2 + 2430(0.01)4 + 1458(0.01)
= 2 + 0.027 + 0.0000243 + 2.0270243
= 2.02702 (5 d.p)
GENERAL EVALUATION
1) Write down and simplify all the terms of the binomial expansion of ( 1 – x )6 . Use the expansion to evaluate 0.9976 correct to 4 dp
2) Write down the expansion of ( 1 + ¼ x ) 5 simplifying all its coefficients
3) Use the binomial theorem to expand ( 2 – ¼ x)5 and simplify all the terms
4) Deduce the expansion of ( 1 – x +x2 )6 in ascending powers of x
Reading Assignment
New Further Maths Project 2 page 73 – 78
WEEKEND ASSIGNMENT
If the first three terms of the expansion of ( 1 + px )n in ascending powers of x are 1 + 20v + 160x find the value of
1) n a) 2 b) 3 c) 4 d) 5
2) p a) 2 b) 3 c) 4 d) 5
3) In the expansion of ( 2x + 3y )4 what is the coefficient of y4 a) 16 b) 81 c) 216 d) 96
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