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Cambridge Maths Academy
P1 §8. The binomial expansion 본문
P1 §8. The binomial expansion
Cambridge Maths Academy 2021. 1. 12. 04:30
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) 4x2−12xy−9y2
(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) 5√x
(b) 1163√x2
(c) 103√x
(d) 16813√x4
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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