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P1 §8. The binomial expansion 본문

A-level Mathematics/Pure Mathematics 1

P1 §8. The binomial expansion

Cambridge Maths Academy 2021. 1. 12. 04:30
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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) (2x3y)2

(b) (xy)3

(c) (2+x)3

 

Answers:

(a) 4x212xy9y2

(b) x3+3x2y+3xy2+y3

(c) 8+12x+6x2+x3

 

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

(a) (2x)3

(b) (3x)4

(c) (25x)2

(d) (13x)3

 

Answers:

(a) 8x3

(b) 181x4

(c) 425x2

(d) 27x3

 

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

(a) (25x)12

(b) (64x)23

(c) (9100x)12

(d) (827x)43

 

Answers:

(a) 5x

(b) 1163x2

(c) 103x

(d) 16813x4

 

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.)

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