Binomial Expansion SS2 Further Mathematics Lesson Note

PASCAL’S TRIANGLE

Consider the expansion of each of the following;

                    (𝑥 + 𝑦)⁰ = 1

(𝑥 + 𝑦)¹ = 1𝑥 + 1𝑦

(𝑥 + 𝑦)² = 1𝑥² + 2𝑥𝑦 + 1𝑦²

(𝑥 + 𝑦)³ = 1𝑥³ + 3𝑥²𝑦 + 3𝑥𝑦² + 1𝑦³

(𝑥 + 𝑦)⁴ = 1𝑥⁴ + 4𝑥³𝑦 + 6𝑥²𝑦² + 4𝑥𝑦³+ 1𝑦⁴

(𝑥 + 𝑦)⁵ = 1𝑥⁵ + 5𝑥⁴𝑦 + 10𝑥³𝑦³ + 10𝑥²𝑦³ + 5𝑥𝑦⁴ + 1𝑦⁵

The coefficients of x and y can be displayed in an array as 

                        1

                      1   1

                      1 2 1

                     1 3 3 1

                    1 4 6 4 1

                 1 5 10 10 5 1

              1 6 15 20 15 6 1

            1 7 21 35 35 21 7 1

The array of coefficients displayed above is called PASCAL’S TRIANGLE and it is used in determining the 

coefficients of the terms of the powers of a binomial expansion

Feature Of The Pascal’s Triangle

• Each line or coefficient is symmetrical
• Each line of coefficients can be obtained from the line of coefficients immediately preceding it
• In the expression of (𝑥 + 𝑦)ⁿ for instant, there are (𝑛 + 1) terms
• In each of the terms, involved in the expansion the power of x and y put together is n
• While the power of x is in decreasing order, the power of y is increasing order

Example 1: By using Pascal’s triangle expand and simplify completely (2𝑥 + 3𝑦)⁴

Solution 

(2𝑥 + 3𝑦)⁴ = (2𝑥)⁴ + 4(2𝑥)³(3𝑦) + 6(2𝑥)²(3𝑦)² + 4(2𝑥)(3𝑦)³ + (3𝑦)⁴

= 16𝑥⁴ + 96𝑥³𝑦 + 216𝑥²𝑦² + 216𝑥𝑦³ + 81𝑦⁴

Example 2: Using Pascal’s triangle, expand and simplify completely (𝑥 − 2𝑦)⁵

Solution 

(𝑥 − 2𝑦)⁵ = 𝑥⁵ + 5𝑥⁴(−2𝑦) + 10𝑥³(−2𝑦)² + 10𝑥²(−2𝑦)³ + 5𝑥(−2𝑦)⁴+(−2𝑦)⁵ = 𝑥⁵ − 10𝑥⁴𝑦 + 40𝑥³𝑦² − 80𝑥²𝑦³ + 80𝑥𝑦⁴ − 32𝑦⁵

Example 3: Using Pascal’s triangle expand and simplify correct to 5 decimal places (1.01)⁴

Solution 

(1.01)4 = (1 + 0.01)⁴ = 1 + 4(0.01) +6(0.01)² + 4(0.01)³ + (0.01)⁴

= 1.04060401

≈ 1.04060(5𝑑. 𝑝)

ASSIGNMENT 

1. Use Pascal’s triangle to expand and simplify completely(𝑥 + 𝑦)
2. Hence, find the coefficients 

of the following; 

(a) 𝑥⁵𝑦⁶

(b) 𝑥𝑦¹⁰

(c) 𝑥⁹𝑦²

(d) 𝑥⁴𝑦⁷

2. What is expansion of (3𝑥 − 𝑦)⁷ by using Pascal’s triangle?
3. Given that (1 + 3𝑥)¹⁰ = 1 + 𝑃𝑥 + 𝑄𝑥² + 𝑅𝑥³ + ⋯, find the values of the integers P, Q and R
4. Find and express in ascending powers of x and with integral coefficients for (1 − 2𝑥)³ −(1 + 2𝑥)³
5. Expand (1 + 𝑥)⁵ in ascending power of x, hence without using any calculating devices, estimate (1.1)⁵, give your answer correct to four decimal places

THE BINOMIAL EXPANSION FORMULA