Lesson 1 of 10 · 7 min
Why a binomial theorem
NCERT §7.1
At the school science fair, the maths stall has a card on its table: work out 98⁵ with no calculator, in under two minutes. Multiplying 98 by itself five times is slow and easy to get wrong. Is there a shortcut?
The story this chapter follows: The school science fair
The lesson in notes
In short
A binomial is a sum of two terms, such as a + b, x − 2 or 2x + 3y. The chapter is about writing out a power of a binomial, (a + b)ⁿ, without multiplying the brackets one at a time.
Squares and cubes are already known: (a + b)² = a² + 2ab + b² and (a + b)³ = a³ + 3a²b + 3ab² + b³. They turn 98² into (100 − 2)² = 10000 − 400 + 4 = 9604 with no long multiplication.
For a higher power such as 98⁵ or 101⁶, repeated multiplication is slow and error-prone. The binomial theorem gives every term of (a + b)ⁿ directly.
The theorem also works for other kinds of index, but this chapter covers only positive integral indices n = 1, 2, 3, ….