Part I: Find (x + 1)6 − (x − 1)6
Step 1: Expand using Binomial Theorem
(x + 1)6 = x6 + 6x5 + 15x4 + 20x3 + 15x2 + 6x + 1
(x − 1)6 = x6 − 6x5 + 15x4 − 20x3 + 15x2 − 6x + 1
Step 2: Subtract
(x + 1)6 − (x − 1)6
= (x6 − x6) + (6x5 + 6x5) + (15x4 − 15x4)
+ (20x3 + 20x3) + (15x2 − 15x2) + (6x + 6x)
= 12x5 + 40x3 + 12x
Result:
(x + 1)6 − (x − 1)6 = 12x5 + 40x3 + 12x
Part II: Evaluate (√2 + 1)6 + (√2 − 1)6
Step 1: Use symmetry
(x + 1)6 + (x − 1)6 = 2(x6 + 15x4 + 15x2 + 1)
Step 2: Substitute x = √2
x2 = 2, x4 = 4, x6 = 8
(x + 1)6 + (x − 1)6
= 2(8 + 15·4 + 15·2 + 1)
= 2(8 + 60 + 30 + 1)
= 2 × 99
= 198
Final Answers:
(x + 1)6 − (x − 1)6 = 12x5 + 40x3 + 12x
(√2 + 1)6 + (√2 − 1)6 = 198