P1 §8. The binomial expansion

Pure mathematics Year 1

Here's a short introduction to Pure maths 1 Chapter 8 The Binomial expansion.

 

8.1 Pascal's triangle: Use Pascal's triangle to identify binomial coefficients and use them to expand simple binomial expressions.

8.2 Factorial notation: Use combinations and factorial notation.

8.3 The binomial expansion: Use the binomial expansion to expand brackets.

8.4 Solving binomial problems: Find individual coefficients in a binomial expansion.

8.5 Binomial estimation: Make approximations using the binomial expansion.

8.6 Mixed exercise for chapter 8

8.7 Review exercise for chapter 8

 

Prior knowledge check

 

Q1. [P1 §1.2 Expanding brackets] Expand and simplify where possible:

(a) $ (2x-3y)^2 $

(b) $ (x-y)^3 $

(c) $ (2+x)^3 $

 

Answers:

(a) $ 4x^2 - 12xy - 9y^2 $

(b) $ x^3 + 3x^2y + 3xy^2 + y^3 $

(c) $ 8 + 12x + 6x^2 + x^3 $

 

Q2. [P1 §1.2 Expanding brackets & P1 §1.4 Negative and fractional indices] Simplify:

(a) $ (-2x)^3 $

(b) $ (3x)^{-4} $

(c) $ \left(\frac25x\right)^2 $

(d) $ \left(\frac13x\right)^{-3} $

 

Answers:

(a) $ -8x^3 $

(b) $ \frac1{81x^4} $

(c) $ \frac{4}{25}x^2 $

(d) $ \frac{27}{x^3} $

 

Q3. [P1 §1.4 Negative and fractional indices] Simplify:

(a) $ (25x)^{\frac12} $

(b) $ (64x)^{-\frac23} $

(c) $ \left(\frac9{100}x\right)^{-\frac12} $

(d) $ \left(\frac8{27}x\right)^{\frac43} $

 

Answers:

(a) $ 5\sqrt{x} $

(b) $ \frac1{16\sqrt[3]{x^2}} $

(c) $ \frac{10}{3\sqrt{x}} $

(d) $ \frac{16}{81}\sqrt[3]{x^4} $

 

Comment: The binomial expansion can be used to expand brackets raised to large powers. It can be used to simplify probability models with a large number of trials, such as those used by manufacturers to predict faults. (See P1 §8.5 Binomial estimation Exercise 8E Q9 and P1 §8.6 Mixed exercise 8 Q3.)