Question

If the coefficient of \(x^{30}\) in the expansion of \(\left(1 + \frac{1}{x}\right)^6 (1 + x^2)^7 (1 - x^3)^8, \, x \neq 0\) is \(\alpha\), then \(|\alpha|\) equals ____.

Answer 1

Correct Answer: 678

We need to find the absolute value of the coefficient of \( x^{30} \) in the expansion of \( \left(1 + \frac{1}{x}\right)^6 (1 + x^2)^7 (1 - x^3)^8 \).

Concept Used:

The solution uses the Binomial Theorem. The general term in the expansion of \( (a+b)^n \) is given by:

\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]

To find the coefficient of a specific power of \( x \) in a product of multiple binomial expansions, we find the general term for each expansion and then combine them. We then solve for the powers that sum to the desired power of \( x \).

Step-by-Step Solution:

Step 1: Rewrite the given expression to isolate the polynomial part.

Let the expression be \( E \).

\[ E = \left(1 + \frac{1}{x}\right)^6 (1 + x^2)^7 (1 - x^3)^8 = \left(\frac{x+1}{x}\right)^6 (1 + x^2)^7 (1 - x^3)^8 \] \[ E = \frac{(1+x)^6}{x^6} (1 + x^2)^7 (1 - x^3)^8 \]

To find the coefficient of \( x^{30} \) in \( E \), we need to find the coefficient of \( x^{30} \times x^6 = x^{36} \) in the polynomial expansion of \( P(x) = (1+x)^6 (1 + x^2)^7 (1 - x^3)^8 \).

Step 2: Write the general term for the product \( P(x) \).

The general term of the expansion of \( P(x) \) is a product of the general terms of each binomial factor:

• For \( (1+x)^6 \), the general term is \( \binom{6}{r_1} x^{r_1} \), where \( 0 \le r_1 \le 6 \).
• For \( (1+x^2)^7 \), the general term is \( \binom{7}{r_2} (x^2)^{r_2} = \binom{7}{r_2} x^{2r_2} \), where \( 0 \le r_2 \le 7 \).
• For \( (1-x^3)^8 \), the general term is \( \binom{8}{r_3} (-x^3)^{r_3} = \binom{8}{r_3} (-1)^{r_3} x^{3r_3} \), where \( 0 \le r_3 \le 8 \).

The general term of \( P(x) \) is the product of these terms:

\[ T = \binom{6}{r_1} \binom{7}{r_2} \binom{8}{r_3} (-1)^{r_3} x^{r_1 + 2r_2 + 3r_3} \]

Step 3: Find integer solutions for the power equation.

We need the power of \( x \) to be 36. So we must solve the linear Diophantine equation:

\[ r_1 + 2r_2 + 3r_3 = 36 \]

subject to the constraints \( 0 \le r_1 \le 6 \), \( 0 \le r_2 \le 7 \), and \( 0 \le r_3 \le 8 \).

We systematically check possible values for \( r_3 \):

• If \( r_3 = 8 \), \( r_1 + 2r_2 = 36 - 24 = 12 \). Possible pairs \( (r_1, r_2) \) are (0, 6), (2, 5), (4, 4), (6, 3).
• If \( r_3 = 7 \), \( r_1 + 2r_2 = 36 - 21 = 15 \). Possible pairs \( (r_1, r_2) \) are (1, 7), (3, 6), (5, 5).
• If \( r_3 = 6 \), \( r_1 + 2r_2 = 36 - 18 = 18 \). Possible pairs \( (r_1, r_2) \) are (4, 7), (6, 6).
• If \( r_3 = 5 \), \( r_1 + 2r_2 = 36 - 15 = 21 \). Since \( 2r_2 \le 14 \), \( r_1 = 21 - 2r_2 \ge 7 \), which is not possible as \( r_1 \le 6 \). Thus, no solutions for \( r_3 \le 5 \).

Step 4: Calculate the coefficient for each valid combination \( (r_1, r_2, r_3) \).

The coefficient for a combination is \( C(r_1, r_2, r_3) = \binom{6}{r_1} \binom{7}{r_2} \binom{8}{r_3} (-1)^{r_3} \).

For \( r_3 = 8 \) (\((-1)^8 = 1\)):

• (0, 6, 8): \( \binom{6}{0}\binom{7}{6}\binom{8}{8}(1) = 1 \cdot 7 \cdot 1 = 7 \)
• (2, 5, 8): \( \binom{6}{2}\binom{7}{5}\binom{8}{8}(1) = 15 \cdot 21 \cdot 1 = 315 \)
• (4, 4, 8): \( \binom{6}{4}\binom{7}{4}\binom{8}{8}(1) = 15 \cdot 35 \cdot 1 = 525 \)
• (6, 3, 8): \( \binom{6}{6}\binom{7}{3}\binom{8}{8}(1) = 1 \cdot 35 \cdot 1 = 35 \)

For \( r_3 = 7 \) (\((-1)^7 = -1\)):

• (1, 7, 7): \( \binom{6}{1}\binom{7}{7}\binom{8}{7}(-1) = 6 \cdot 1 \cdot 8 \cdot (-1) = -48 \)
• (3, 6, 7): \( \binom{6}{3}\binom{7}{6}\binom{8}{7}(-1) = 20 \cdot 7 \cdot 8 \cdot (-1) = -1120 \)
• (5, 5, 7): \( \binom{6}{5}\binom{7}{5}\binom{8}{7}(-1) = 6 \cdot 21 \cdot 8 \cdot (-1) = -1008 \)

For \( r_3 = 6 \) (\((-1)^6 = 1\)):

• (4, 7, 6): \( \binom{6}{4}\binom{7}{7}\binom{8}{6}(1) = 15 \cdot 1 \cdot 28 = 420 \)
• (6, 6, 6): \( \binom{6}{6}\binom{7}{6}\binom{8}{6}(1) = 1 \cdot 7 \cdot 28 = 196 \)

Final Computation & Result:

Step 5: Sum all the calculated coefficients to find the total coefficient \( \alpha \).

\[ \alpha = (7 + 315 + 525 + 35) + (-48 - 1120 - 1008) + (420 + 196) \] \[ \alpha = 882 - 2176 + 616 \] \[ \alpha = 1498 - 2176 = -678 \]

The question asks for the value of \( |\alpha| \).

\[ |\alpha| = |-678| = 678 \]

The value of \( |\alpha| \) is 678.