Understanding the Coefficient of x^7 in Polynomial Expansion Using the Multinomial Theorem

When dealing with polynomial expansions, understanding the coefficients of certain terms requires a clear grasp of the multinomial theorem. This article will explore how to find the coefficient of x7 in the polynomial expression 1 - 2x x24. We will use the multinomial theorem to solve the problem step-by-step.

Introduction to the Multinomial Theorem

The multinomial theorem is a generalization of the binomial theorem, which allows us to expand expressions of the form abc^n. According to the theorem, the coefficient of a^{n_1}b^{n_2}c^{n_3} in the expansion of abc^n is given by:

(frac{n!}{n_1! n_2! n_3!})

Where n_1, n_2, n_3 in mathbb{N} and n_1 n_2 n_3 n.

Applying the Multinomial Theorem to the Given Polynomial

Consider the polynomial expression 1 - 2x x^2^4. We aim to find the coefficient of x^7 in its expansion. To do this, we will use the multinomial expansion theorem on the polynomial 1 - (2x) (x^2)^4, where a 1, b -2x, c x^2, and n 4.

Setting Up the Expansion

The multinomial expansion is given by:

(sum_{i j kn} frac{n!}{i!j!k!} a^i b^j c^k)

Here, we need to determine the values of i, j, and k such that:

0i j 2k 7 i j k 4

Solving the Equations

From the second equation, we can express i as:

(i 4 - j - k)

Substituting this into the first equation:

(j 2(4 - j - k) 7)

Expanding and simplifying:

(j 8 - 2j - 2k 7)
(-j - 2k -1)
(j 2k 1)

Now we need to find integer values of j and k that satisfy both j 2k 1 and 0 leq i, j, k leq 4.

Checking Solutions

Let's test the possible values for k:

(k 0) : (j 2(0) 1) lt; Not possible since (j) cannot be negative. (k 1) : (j 2(1) 1) lt; No valid (j) since (j) must be 1 and not a positive integer. (k 2) : (j 2(2) 1) lt; No valid solution. (k 3) : (j 2(3) 1) lt; No valid solution. (k 4) : (j 2(4) 1) lt; Not possible since (j) cannot be negative.

The only valid combination of (j 1), (k 3), and consequently (i 0), fits our requirement.

Calculating the Coefficient

The multinomial coefficient for the valid combination is:

(frac{4!}{0! 1! 3!} frac{24}{1 cdot 1 cdot 6} 4)

The term corresponding to these values is:

1^0 cdot (-2x)^1 cdot (x^2)^3 -2x cdot x^6 -2x^7)

Thus, the coefficient is:

4 cdot (-2) -8)

Conclusion

The coefficient of x^7 in the expansion of 1 - 2x x^2^4 is -8. This is the result of using the multinomial theorem effectively to solve the problem step-by-step.