Step 1: Understanding the Concept:
We can simplify the expression inside the bracket before expanding. : Key Formula or Approach:
Binomial theorem for negative index: \( (1+x)^{-n} = 1 - nx + \frac{n(n+1)}{2!}x^2 - \dots \). Step 2: Detailed Explanation:
The expression is \( (1 + x + x^2 + x^3)^{-3} \).
Factorize the cubic polynomial:
\[ 1 + x + x^2 + x^3 = (1 + x) + x^2(1 + x) = (1 + x)(1 + x^2) \]
So the expression is:
\[ [ (1 + x)(1 + x^2) ]^{-3} = (1 + x)^{-3} (1 + x^2)^{-3} \]
Now, expand both using the binomial series:
\[ (1 + x)^{-3} = 1 - 3x + 6x^2 - \dots \]
\[ (1 + x^2)^{-3} = 1 - 3x^2 + 6x^4 - \dots \]
Multiply the two series:
\[ \text{Product} = (1 - 3x + 6x^2 - \dots)(1 - 3x^2 + \dots) \]
To find the coefficient of \( x \), we look for terms whose multiplication results in \( x^1 \):
- \( 1 \times (\text{no } x \text{ term in second series}) \)
- \( (-3x) \times (1) = -3x \)
No other combination yields \( x \).
Thus, the coefficient is -3. Step 3: Final Answer:
The coefficient of x is -3.