Without expanding, find the value of (x + 1)4 − 4(x + 1)3 (x − 1) + 6 (x + 1)2 (x − 1)2 − 4(x + 1) (x − 1)3 + (x − 1)4 - Mathematics and Statistics

Sum

Without expanding, find the value of (x + 1)⁴ − 4(x + 1)³ (x − 1) + 6 (x + 1)² (x − 1)² − 4(x + 1) (x − 1)³ + (x − 1)⁴

Let x + 1 = a and x − 1 = b

We notice that 1, 4, 6, 4, 1 are the values of ⁴C₀, ⁴C₁, ⁴C₂, ⁴C₃, ⁴C₄ respectively and

∴ the given expression becomes

⁴C₀a⁴b⁰ − ⁴C₁a³b + ⁴C₂a²b² − ⁴C₃ab³ + ⁴C₄a⁰b⁴

= (a − b)⁴

= [x + 1 − (x − 1)]⁴

= (x + 1 − x + 1)⁴

= 2⁴

= 16

Concept: Binomial Theorem for Positive Integral Index

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Chapter 4 Methods of Induction and Binomial Theorem
Exercise 4.2 | Q 7. (i) | Page 77