Chapter 8 - The Binomial Expansion

Question 1

How is Pascal’s triangle formed?

Answer 1

It is formed by adding adjacent pairs of numbers to find the numbers on the next row.

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

Question 2

What is the row of Pascal’s triangle that gives you the correct coefficients in the expansion of (a + b)ⁿ ?

Answer 2

It is the (n + 1)th row.

1st row (a + b)⁰

2nd row (a + b)¹

3rd row (a + b)²

…

Question 3

What is n! equal to?

Answer 3

n x (n - 1) x (n - 2) x (n - 3) x … x 1.

so 6! = 6 x 5 x 4 x 3 x 2 x 1

Question 4

What are the two forms of factorial notation?

Answer 4

ⁿC_r and

( n )

( r )

They both mean:

n

r! (n - r)!

Question 5

What is the general term in the binomial expansion of (a + b)ⁿ?

Answer 5

ⁿC_r a^{n - r} b^r

Question 6

How do you use the binomial expansion to estimate a value? Example: Use the binomial expansion of (1 - x/4)¹⁰ to estimate the value of 0.975¹⁰?

Answer 6

• Use the binomial expansion to expand (1 - x/4)¹⁰ for however many terms as it asks you to in the question eg. 4 terms gives you 1 - 2.5x + 2.8125x² - 1.875x³ + ……
• Set what is inside the bracket equal to what you are estimating: 1 - x/4 = 0.975 and solve to find x. x = 0.1
• Substitute x = 0.1 into the first four terms of the expansion above and work out an estimate value. eg (0.975)¹⁰ estimated to be 1 - 2.5(0.1) + 2.8125 (0.1)² - 1.875 (0.1)³ = 0.7763