Binomial TheoremMedium
Question
The expression [x + (x3 -1)1/2]5 + [x - (x3 -1)1/2]5 is a polynomial of degree
Options
A.5
B.7
C.6
D.8
Solution
We know that
(a + b)5 + (a - b)5 = 5C0a5 + 5C1a4b + 5C2a3b2
+ 5C3a2b3 + 5C4ab5 + 5C5b5 + 5C0a5- 5C1a4b
+ 5C2a3b2 - 5C3a2b3 + 5C4ab4 - 5C5b5
= 2[a5 + 10a3b2 + 5ab4]
∴ [x + (x3 - 1)1/2]5 + [x - (x3 - 1)1/2]5
= 2[x5 + 10x3 (x3 - 1) + 5x(x3 -1)2]
Therefore, the given expression is a polynomial of degree 7.
(a + b)5 + (a - b)5 = 5C0a5 + 5C1a4b + 5C2a3b2
+ 5C3a2b3 + 5C4ab5 + 5C5b5 + 5C0a5- 5C1a4b
+ 5C2a3b2 - 5C3a2b3 + 5C4ab4 - 5C5b5
= 2[a5 + 10a3b2 + 5ab4]
∴ [x + (x3 - 1)1/2]5 + [x - (x3 - 1)1/2]5
= 2[x5 + 10x3 (x3 - 1) + 5x(x3 -1)2]
Therefore, the given expression is a polynomial of degree 7.
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