Question

The coefficient of \(x\) in the expansion of \((1 + x + x^2 + x^3)^{-3\) is}

• A. 6
• B. 9
• C. 5
• D. -3

Hint:

To find coefficient of low powers, ignore higher powers like \(u^2, u^3\) if they cannot produce that term.

Answer 1

The Correct Option is D

Concept: \[ (1+u)^{-3} = 1 - 3u + 6u^2 - 10u^3 + \dots \]

Step 1: Substitution. Let \(u = x + x^2 + x^3\).

Step 2: Identify terms contributing to \(x\).
• From \(1\) → no \(x\)
• From \(-3u\): \(-3(x + x^2 + x^3)\) → contributes \(-3x\)
• From \(6u^2\): \(u^2\) has minimum power \(x^2\) → no \(x\) term
• Higher powers give \(x^3\) or more

Step 3: Conclusion. Only \(-3u\) contributes to \(x\). \[ \text{Coefficient of } x = -3 \]