Expanding Binomial Expressions: A Step-by-Step Guide Using the Binomial Theorem

In this article, we will explore the process of expanding binomial expressions using the binomial and multinomial theorems. We will use these theorems to expand a specific expression, namely

(1 -u00bdx - x^2)^4

in ascending powers of x up to the term x^3. We will break down the process into easy-to-follow steps and highlight the key concepts involved.

Understanding the Binomial Theorem

The binomial theorem is a powerful mathematical tool used to expand expressions of the form (a b)^n. It states that:

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

where binom{n}{k} is the binomial coefficient. This theorem is particularly useful when dealing with binomial expressions. However, for more complex cases, we might need to use the multinomial theorem.

Introducing the Multinomial Theorem

The multinomial theorem generalizes the binomial theorem to expressions with more than two terms. For an expression of the form (x_1 x_2 ... x_m)^n, the multinomial theorem states:

(x_1 x_2 ... x_m)^n sum_{k_1 k_2 ... k_m n} frac{n!}{k_1! k_2! ... k_m!} x_1^{k_1} x_2^{k_2} ... x_m^{k_m}

Here, the sum is over all non-negative integer solutions to the equation k_1 k_2 ... k_m n, and frac{n!}{k_1! k_2! ... k_m!} is the multinomial coefficient.

Step-by-Step Expansion of (1 -u00bdx - x^2)^4

Let's now proceed with expanding the given expression (1 -u00bdx - x^2)^4 using the multinomial theorem. We will break down the process into several steps.

Step 1: Identifying the Terms

The terms in our expression are:

a 1 b -u00bdx c -x^2

We want to expand (1 -u00bdx - x^2)^4.

Step 2: Applying the Multinomial Theorem

According to the multinomial theorem, we need to find the multinomial coefficients and the corresponding terms for the expansion. The equation we need to solve is:

k_1 k_2 k_3 4

We need to ensure that the total degree in x does not exceed 3. The possible combinations are:

k_3 0 k_3 1 k_3 2 k_3 3 k_3 4

However, k_3 4 is impossible as it would result in x^8 which is higher than x^3.

Step 3: Calculating the Terms

We will calculate the terms for each valid combination of k_2 and k_3:

k_3 0: k_1 k_2 4 k_3 1: k_1 k_2 3 k_3 2: k_1 k_2 2 k_3 3: k_1 k_2 1

For each combination, we will calculate the relevant terms and coefficients.

Case 1: k_3 0

k_1 4, k_2 0

Term: 1^4

Coefficient: binom{4}{4,0,0} 1

Contribution: 1

Case 2: k_3 1

k_1 3, k_2 1

Term: 1^3 * (-u00bdx) * (-x^2)^1

Term: -u00bdx * -x^2 u00bdx^3

Coefficient: binom{4}{3,1,0} 4

Contribution: 4 * u00bdx^3 2x^3

Case 3: k_3 2

k_1 2, k_2 2

Term: 1^2 * (-u00bdx)^2 * (-x^2)^2

Term: (u00bdx^2) * (x^4) u00bdx^6

Coefficient: binom{4}{2,2,0} 6

Contribution: 6 * u00bdx^2 3x^2

Case 4: k_3 3

k_1 1, k_2 1

Term: 1^1 * (-u00bdx) * (-x^2)^3

Term: -u00bdx * (-x^6) u00bdx^7

Not included as x^7 exceeds x^3

Case 5: k_3 4

Not possible as k_1 k_2 0

Step 4: Combining All Contributions

Combining all the contributions, we get:

1 2x^3 u00bdx^2

However, we only need the terms up to x^3, so we can rewrite the final result as:

1 2x u00bdx^2

Final Result

Thus, the expansion of (1 -u00bdx - x^2)^4 in ascending powers of x up to the term x^3 is: