The Binomial Expansion
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The Binomial Expansion. Introduction. You first met the Binomial Expansion in C2 In this chapter you will have a brief reminder of expanding for positive integer powers We will also look at how to multiply out a bracket with a fractional or negative power

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The Binomial Expansion

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The binomial expansion

The Binomial Expansion


Introduction

Introduction

  • You first met the Binomial Expansion in C2

  • In this chapter you will have a brief reminder of expanding for positive integer powers

  • We will also look at how to multiply out a bracket with a fractional or negative power

  • We will also use partial fractions to allow the expansion of more complicated expressions


The binomial expansion

Teachings for Exercise 3A


The binomial expansion

The Binomial Expansion

You need to be able to expand expressions of the form (1 + x)n where n is any real number

Find:

Always start by writing out the general form

Sub in:

n = 4

x = x

Work out each term separately and simplify

Every term after this one will contain a (0) so can be ignored

 The expansion is finite and exact

3A


The binomial expansion1

The Binomial Expansion

You need to be able to expand expressions of the form (1 + x)n where n is any real number

Find:

Always start by writing out the general form

Sub in:

n = 3

x = -2x

Work out each term separately and simplify

It is VERY important to put brackets around the x parts

Every term after this one will contain a (0) so can be ignored

 The expansion is finite and exact

3A


The binomial expansion2

The Binomial Expansion

You need to be able to expand expressions of the form (1 + x)n where n is any real number

Find:

Rewrite this as a power of x first

Write out the general form (it is very unlikely you will have to go beyond the first 4 terms)

Sub in:

n = -1

x = x

Work out each term separately and simplify

  • With a negative power you will not get a (0) term

  • The expansion is infinite

  • It can be used as an approximation for the original term

3A


The binomial expansion3

The Binomial Expansion

You need to be able to expand expressions of the form (1 + x)n where n is any real number

Find:

Rewrite this as a power of x first

Write out the general form (it is very unlikely you will have to go beyond the first 4 terms)

Sub in:

n = 1/2

x = -3x

Work out each term separately and simplify

 You should use your calculator carefully

  • With a fractional power you will not get a (0) term

  • The expansion is infinite

  • It can be used as an approximation for the original term

3A


The binomial expansion4

The Binomial Expansion

You need to be able to expand expressions of the form (1 + x)n where n is any real number

and state the values of x for which it is valid…

Find the Binomial expansion of:

Write out the general form

Sub in:

n = 1/3

x = -x

Work out each term separately and simplify

Imagine we substitute x = 2 into the expansion

The values fluctuate (easier to see as decimals)

 The result is that the sequence will not converge and hence for x = 2, the expansion is not valid

3A


The binomial expansion5

The Binomial Expansion

You need to be able to expand expressions of the form (1 + x)n where n is any real number

and state the values of x for which it is valid…

Find the Binomial expansion of:

Write out the general form

Sub in:

n = 1/3

x = -x

Work out each term separately and simplify

Imagine we substitute x = 0.5 into the expansion

The values continuously get smaller

 This means the sequence will converge (like an infinite series) and hence for x = 0.5, the sequence IS valid…

27

3A


The binomial expansion6

The Binomial Expansion

You need to be able to expand expressions of the form (1 + x)n where n is any real number

and state the values of x for which it is valid…

Find the Binomial expansion of:

Write out the general form

Sub in:

n = 1/3

x = -x

Work out each term separately and simplify

How do we work out for what set of values x is valid?

The reason an expansion diverges or converges is down to the x term…

If the term is bigger than 1 or less than -1, squaring/cubing etc will accelerate the size of the term, diverging the sequence

If the term is between 1 and -1, squaring and cubing cause the terms to become increasingly small, to the sum of the sequence will converge, and be valid

Write using Modulus

The expansion is valid when the modulus value of x is less than 1

3A


The binomial expansion7

The Binomial Expansion

You need to be able to expand expressions of the form (1 + x)n where n is any real number

and state the values of x for which it is valid…

Find the Binomial expansion of:

Write out the general form:

Sub in:

n = -2

x = 4x

Work out each term separately and simplify

The ‘x’ term is 4x…

Divide by 4

3A


The binomial expansion8

The Binomial Expansion

You need to be able to expand expressions of the form (1 + x)n where n is any real number

and by using x = 0.01, find an estimate for √2

Find the Binomial expansion of:

Write out the general form:

Sub in:

n = 1/2

x = -2x

Work out each term separately and simplify

3A


The binomial expansion9

The Binomial Expansion

You need to be able to expand expressions of the form (1 + x)n where n is any real number

and by using x = 0.01, find an estimate for √2

Find the Binomial expansion of:

x = 0.01

Rewrite left using a fraction

Square root top and bottom separately

Multiply by 10

Divide by 7

3A


The binomial expansion

Teachings for Exercise 3B


The binomial expansion10

The Binomial Expansion

You can use the expansion for (1 + x)n to expand (a + bx)n by taking out a as a factor

Find the first 4 terms in the Binomial expansion of:

Take a factor 4 out of the brackets

Both parts in the square brackets are to the power 1/2

You can work out the part outside the bracket

Write out the general form:

Sub in:

n = 1/2

x = x/4

Work out each term carefully and simplify it

Remember we had a 2 outside the bracket

 Multiply each term by 2

Multiply by 4

3B


The binomial expansion11

The Binomial Expansion

You can use the expansion for (1 + x)n to expand (a + bx)n by taking out a as a factor

Find the first 4 terms in the Binomial expansion of:

Take a factor 2 out of the brackets

Both parts in the square brackets are to the power -2

You can work out the part outside the bracket

Write out the general form:

Sub in:

n = -2

x = 3x/2

Work out each term carefully and simplify it

Remember we had a 1/4 outside the bracket

 Divide each term by 4

Multiply by 2, divide by 3

3B


The binomial expansion

Teachings for Exercise 3C


The binomial expansion12

The Binomial Expansion

You can use Partial fractions to simplify the expansions of more difficult expressions

Find the expansion of:

up to and including the term in x3

Express as Partial Fractions

Cross-multiply and combine

The numerators must be equal

If x = 2

If x = -1

Express the original fraction as Partial Fractions, using A and B

3C


The binomial expansion13

The Binomial Expansion

You can use Partial fractions to simplify the expansions of more difficult expressions

Find the expansion of:

up to and including the term in x3

Both fractions can be rewritten

Expand each term separately

Write out the general form:

Sub in:

x = x

n = -1

Work out each term carefully

Remember that this expansion is to be multiplied by 3

3C


The binomial expansion14

The Binomial Expansion

You can use Partial fractions to simplify the expansions of more difficult expressions

Find the expansion of:

up to and including the term in x3

Both fractions can be rewritten

Expand each term separately

Take a factor 2 out of the brackets (and keep the current 2 separate…)

Both parts in the square brackets are raised to -1

Work out 2-1

This is actually now cancelled by the 2 outside the square bracket!

3C


The binomial expansion15

The Binomial Expansion

You can use Partial fractions to simplify the expansions of more difficult expressions

Find the expansion of:

up to and including the term in x3

Both fractions can be rewritten

Expand each term separately

Write out the general form:

Sub in:

x = -x/2

n = -1

Work out each term carefully

3C


The binomial expansion16

The Binomial Expansion

You can use Partial fractions to simplify the expansions of more difficult expressions

Find the expansion of:

up to and including the term in x3

Both fractions can be rewritten

Replace each bracket with its expansion

Subtract the second from the first (be wary of double negatives in some questions)

3C


Summary

Summary

  • We have been reminded of the Binomial Expansion

  • We have seen that when the power is a positive integer, the expansion is finite and exact

  • With negative or fractional powers, the expansion is infinite

  • We have seen how to decide what set of x-values the expansion is valid for

  • We have also used partial fractions to break up more complex expansions


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