Understanding the 6th Term of the Binomial Expansion for 4 - x^6

In this article, we delve into the process of determining the 6th term of the expansion of 4 - x^6 using the Binomial Theorem. The Binomial Theorem is a powerful mathematical tool that allows us to find specific terms in a binomial expansion without having to expand the entire expression. We will break down the process step-by-step and provide a detailed explanation, along with a step-by-step calculation.

The Binomial Theorem and Its Application

The Binomial Theorem states that for any non-negative integer n and any real numbers a and b, the expansion of a b^n is given by:

a b^n Sum; (k0)^n binom{n}{k} a^(n-k) b^k

Applying the Theorem to 4 - x^6

Let's apply the Binomial Theorem to the expression 4 - x^6. For this, we let:

a 4 b -x n 6

The general term in the binomial expansion is given by:

T_k binom{n}{k} a^(n-k) b^k

For the 6th term, we need to find T_6, which corresponds to k 5. Therefore:

Calculating the 6th Term

Calculate binom{6}{5}

binom{6}{5} 6

Calculate 4^(6-5)

4^1 4

Calculate -x^5

-x^5 -x^5

Substituting these values into the expression for T_6:

T_6 6 4 -x^5 -24x^5

Thus, the 6th term of the expansion of 4 - x^6 is:

boxed{-24x^5}

Alternative Approach: Direct Expansion

A direct way to write the expansion of 4 - x^6 is by using the binomial expansion formula:

4 - x^6 Sum; (m0)^6 binom{6}{m} 4^(6-m) (-x)^m

For the 6th term, we set m 5 to get:

binom{6}{5} 4^(6-5) (-x)^5 binom{6}{5} 4^1 -x^5 6 4 -x^5 -24x^5

Visualization Through Pascal's Triangle

To provide further insight, we can visualize the coefficients using Pascal's Triangle. The coefficients for the expansion of x y^6 are derived as follows:

0 1 2 3 4 5 6 0 1 1 1 1 1 1 1 1 2 1 1 1 1 2 1 3 3 1 1 1 3 1 4 6 4 1 1 4 1 5 10 10 5 1 5 1 6 15 20 15 6 1 6 1 7 21 35 35 21 7 7 1 8 28 56 70 56 8

The coefficients for the expansion of x y^6 are: 1, 6, 15, 20, 15, 6, and 1. For the 6th term, the coefficient is 6. Therefore, we have:

64 - 6144x 384x^2 - 128x^3 24x^4 - 24x^5 x^6

So, the 6th term is:

64 - 6144x 384x^2 - 128x^3 24x^4 - 24x^5 x^6

Thus, the term involving x^5 is:

emdash;24x^5