SS2 First Term Further Mathematics Lesson Note – Binomial Expansion

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WEEK FIVE

TOPIC: BINOMIAL EXPANSION 1

SUB-TOPICS:

• Pascal triangle.
• Binomial expansion of (a+b)ⁿ , where n can be positive integer, negative integer or fractional value.

SUB-TOPIC ONE

The Pascal’s triangle is a format for getting the coefficients of expansions. It applies to binomial and binomial formed from a reduced polynomial.

Consider the expansion of each of the following:

(a+b)⁰

(a+b)¹

(a+b)²

(a+b)³

(a+b)⁴

(a+b)⁵

By ordinary expansion of algebraic terms, we have:

(a+b)⁰ = 1

(a+b)¹ = 1a + 1b

(a+b)² = 1a²+2ab+1b²

(a+b)³ = 1a³+3a²b+3ab²+1b³

(a+b)⁴ = 1a⁴+4a³b+6a²b²+4ab³+1b⁴

(a+b)⁵ = 1a⁵+5a⁴b+10a³b²+10a²b³+5ab⁴+1b⁵

Consider the array of coefficients of a and b. We can display it as follows:

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