AP EAMCET · Maths · Binomial Theorem
The coefficient of \(x^3\) in the expansion of \(\frac{x^4+1}{\left(x^2+1\right)(x-1)}\) when it is expressed in terms of positive integral powers of \(x\), is
- A 0
- B 1
- C 16
- D 24
Answer & Solution
Correct Answer
(A) 0
Step-by-step Solution
Detailed explanation
\(\frac{x^4+1}{\left(x^2+1\right)(x-1)} = x+1 + \frac{2}{\left(x^2+1\right)(x-1)}\) \(= x+1 + \frac{2}{-(1-x)(1+x^2)}\) \(= x+1 - 2(1-x)^{-1}(1+x^2)^{-1}\) \(= x+1 - 2(1+x+x^2+x^3+\dots)(1-x^2+x^4-\dots)\) Coefficient of \(x^3\) in \((1+x+x^2+x^3+\dots)(1-x^2+x^4-\dots)\):…
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