C4 – Algebra(Ch 7): Binomial Expansion
Example of a past paper question (June 12):
1) Why is this a Binomial Expansion question?
2) Do you spot something which is new
compared to C1?
C4 – Algebra(Ch 7): Binomial Expansion
Crucial point to be successful in this topic:
– Binomial expansion knowledge from C1
– Great care required when dealing with fractions,
signs
– Make sure that you know the validity of your
expansion
C4 – Algebra(Ch 7): Binomial Expansion
In AS we dealt with the binomial expansions with n
In A2 we deal with binomial expansions with n is any real number!
....BUT – there are two important differences to note
when n
:
•The series is infinite (i.e. Non-terminating) and we
only use a certain number of terms as approximation
•The expansion of (1+x)n is only valid if
C4 – Algebra(Ch 7): Binomial Expansion
Example 1:
1) Expand (1 + x) -3 up to the term x3.
2) state the values of x for which the expansion is valid.
-> answer
C4 – Algebra(Ch 7): Binomial Expansion
Example:
1) Expand (1 + x) -3 up to the term x3.
2) state the values of x for which the expansion is valid.
Solution:
1)
2)
C4 – Algebra(Ch 7): Binomial Expansion
Example 2:
1) Find the first four terms in the binomial expansion
2) State the set of values of x for which the expansion is
valid.
C4 – Algebra(Ch 7): Binomial Expansion
Example 2:
1) Find the first four terms in the binomial expansion
2) State the set of values of x for which the expansion is
valid.
Ex 7A: Q1 :a – c on examples i) – iv), Q2
C4 – Algebra(Ch 7): Binomial Expansion
What’s the problem ?:
Find the first four terms in the binomial expansion (4 + x)1/2
C4 – Algebra(Ch 7): Binomial Expansion
Sometimes we need to use a Binomial Expansion to
approximate expressions like (a + x)n .
In order to use our formula we need to take a outside the
brackets:
This expansion is hence valid for
C4 – Algebra(Ch 7): Binomial Expansion
Example Binomial Expansion of form (a + x)n:
Find the first four terms in the binomial expansion
Valid for ?
C4 – Algebra(Ch 7): Binomial Expansion
Why bother?
•Historically, the Binomial series has been used to approximate
calculations w/o calculator – the more terms you calculate the
better the approximation.(try √2 as (1+1)^1/2)
•A series can only be used to approximate if the terms decrease
•When І x І < 1, the magnitude of x, x2, x3, x4, ... Form a decreasing
geometric sequence
•Hence, the expansion of (1+x)n is only valid if І x І < 1.
Homework (due Tue, 24/09):
Ex 7A: Q1 :a – d on examples i) – iv) + Q3,4, 5*, 6*
-> warning: your homework may include questions like
Use ICT resources:
•DESMOS to plot the graph
• Wolframalpha to get more terms of the series and see
how it changes the approximation
This is homework for two lessons (there is no lesson on
Friday 20th!)