C2: Chapter 5 Binomial Expansion

Dr J Frost (jfrost@tiffin.kingston.sch.uk)

Last modified: 22nd September 2013

Starter

a) Expand (a + b)0

b) Expand (a + b)1

c) Expand (a + b)2

d) Expand (a + b)3

e) Expand (a + b)4

11a + 1b

1a2 + 2ab + b2

1a3 + 3a2b + 3ab2 + 1b3

1a4 + 4a3b + 6a2b2 + 4ab3 + b4

What do you notice about:

The coefficients: They follow Pascal’s triangle.The powers of a and b: Power of a decreases each time (starting at the power)

Power of b increases each time (starting at 0)

?????

??

Quickfire PascalWhat coefficients in your expansion do you use if the power is:

2: 1 2 1 ?

Quickfire PascalWhat coefficients in your expansion do you use if the power is:

4: 1 4 6 4 1?

Quickfire PascalWhat coefficients in your expansion do you use if the power is:

3: 1 3 3 1?

Quickfire PascalWhat coefficients in your expansion do you use if the power is:

5: 1 5 10 10 5 1?

Quickfire PascalWhat coefficients in your expansion do you use if the power is:

2: 1 2 1 ?

Quickfire PascalWhat coefficients in your expansion do you use if the power is:

4: 1 4 6 4 1?

Quickfire PascalWhat coefficients in your expansion do you use if the power is:

3: 1 3 3 1?

Quickfire PascalWhat coefficients in your expansion do you use if the power is:

5: 1 5 10 10 5 1?

Quickfire PascalWhat coefficients in your expansion do you use if the power is:

4: 1 4 6 4 1?

Binomial Expansion

(x + 2y)4 =

x4 + x3 + x2 + x + (2y) (2y)2 (2y)3 (2y)4

1 4 6 4 1

Step 1: You could first put in the first term with decreasing powers.

Step 2: Put in your second term with increasing powers, starting from 0 (i.e. so that 2y doesn’t appear in the first term of the expansion, because the power is 0)

Step 3: Add the coefficients according to Pascal’s Triangle.

= x4 + 8x3y + 24x2y2 + 32xy3 + 16y4

Your go...

(2x – 5)3

= (2x)3 + 3(2x)2(-5) + 3(2x)(-5)2 + (-5)3

= 8x3 – 60x2 + 150x – 125 ?

Bro Tip: If one of the terms in the bracket is negative, the terms in the result will oscillate between positive and negative.

The coefficient of x2 in the expansion of (2 – cx)3 is 294. Find the possible value(s) of the constant c.

(2 – cx)3 = 23 + 3 22(-cx) + 3 21(-cx)2 + ...So coefficient of x2 is 6c2 = 294c = 7

Bro Tip: When asked about a particular term, it’s helpful to write out the first few terms of the expansion, until you write up to the one needed. There’s no point of simplifying the whole expansion!

?

Exercises

Page 79 Exercise 5A1c, d, g, h2d, g3, 4, 6

How are the rows of Pascal’s Triangle generated?

How many ways are there of choosing 0 items from 4?= 4C0 =

How many ways are there of choosing 1 item from 4?= 4C1 =

How many ways are there of choosing 2 items from 4?= 4C2 =

How many ways are there of choosing 3 items from 4?= 4C3 =

How many ways are there of choosing 4 items from 4?= 4C4 =

?

?

?

?

?

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

Binomial Coefficients

This is known as a binomial coefficient. It can also be written as nCr (said: “n choose r”)

?

?

?

?

?

? ? ?

Binomial CoefficientsTo calculate Binomial Coefficients easily:

Because when we divide 8! by 6!, we cancel out all the numbers between 1 and 6 in the product.i.e. The bottom number of the binomial coefficient (2) tells us how many consecutive numbers we multiply together.

?

?

?

?

?

General formula

Edexcel May 2013 (Retracted)

?

Edexcel Jan 2012

Using Binomial Expansions for approximations

If , then x = 0.1. Plugging this in to our expansion:1 + 0.2 + 0.0175 + 0.00875 = 1.218375Actual value is (1.025)8 = 1.218403. So it is correct to 4dp!

?

Using Binomial Expansions for approximationsExercise 5C

Q8 Write down the first four terms in the expansion of By substituting an appropriate value of x, find an approximate value to (2.1)10. Use your calculator to determine your approximation’s degree of accuracy.

1024 + 1024x + 460.8x2 + 122.88x3

1666.56, which is accurate to 3sf

Q7 Write down the first four terms in the expansion of By substituting an appropriate value for x, find an approximate value to (0.99)6. Use your calculator to determine the degree of accuracy of your approximation.

1 – 0.6x + 0.15x2 – 0.02x3

0.94148, which is accurate to 5dp ?

?

Using Binomial Expansions for approximations

Edexcel January 2007

a) 1 + 5(-2x) + 10(-2x)2 + 10(-2x)3

= 1 – 10x + 40x2 – 80x3

b) We discard the x2 and x3 terms above.(1+x)(1-10x) = 1 – 10x + x – 10x2 = 1 – 9x – 10x2

1- 9x (since we can discard the x2 term again)

?

?