Pure Chapter 8 – The Binomial Expansion
8.3 – The Binomial Expansion
What is the formula for the binomial expansion?
Example
Find the first 4 terms in the expansion of (1 + 2𝑥)8
Example
Find the first four terms in the expansion of (2 − 3𝑥)10
Try these… Exercise 8B Q4a, d and g
Use the binomial expansion to find the first four terms, in ascending powers of 𝑥, of:
a)
(1 + 𝑥)8
d)
(1 − 3𝑥)5
g)
(2 − 3𝑥)6
Exam Question
a) Find the first three terms, in ascending powers of 𝑥, of the binomial expansion
(2 − 3𝑥)5 giving each term in its simplest form
b) Hence, or otherwise, find the first three terms, in ascending powers of 𝑥, of the
expansion of 1 +
𝑥
3
(2 − 3𝑥)5
8.4 Solving Binomial Problems
What formula can we use for a specific term in the expansion?
Example
a) Find the coefficient of 𝑥 4 in the expansion of (1 + 𝑥)6
b)
Find the coefficient of 𝑥 7 in the expansion of (3 − 𝑥)15
Try these…
Find the coefficient of the 𝑥 4 term in the expansion of
(1 − 2𝑥)8
Find the coefficient of the 𝑥 3 term in the expansion of
(4 − 𝑥)7
Exercise 8D Question 2
The coefficient of 𝑥 2 in the expansion of (2 + 𝑎𝑥)6 is 60. Find two possible values of the
constant 𝑎.
Exam Question
a) Find the first three terms, in ascending powers of 𝑥, of the binomial expansion of
(3 − 2𝑥)5 giving each term in its simplest form.
b)
c)
The expansion, in ascending powers of 𝑥, of
𝑓 𝑥 = 1 + 𝑘𝑥 3 − 2𝑥 5
where 𝑘 is a constant, up to and including the term in 𝑥 2 is
𝐴 − 324𝑥 + 𝐵𝑥 2
Find the value of 𝑘.
Hence find the values of 𝐴 and 𝐵