(a+b+c+d)^2 Expansion

4 min read Jul 03, 2024

(a + b + c + d)^2 Expansion

The expansion of (a + b + c + d)^2 is a fundamental concept in algebra and is used extensively in various mathematical and scientific applications. In this article, we will explore the step-by-step process of expanding this expression.

Binomial Theorem

The binomial theorem is a powerful tool for expanding powers of binomials, which are expressions of the form (a + b). The theorem states that:

$(a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k$

where n is a positive integer, and {n \choose k} is the binomial coefficient.

Cubing the Expression

To expand (a + b + c + d)^2, we can use the binomial theorem with n = 2. However, since we have four variables, we need to apply the theorem twice.

Let's start by rewriting the expression as:

$(a + b + c + d)^2 = ((a + b) + (c + d))^2$

Now, we can apply the binomial theorem to the inner expression:

$(a + b + c + d)^2 = ((a + b) + (c + d))^2 = (a + b)^2 + 2(a + b)(c + d) + (c + d)^2$

Expanding the Squares

Next, we need to expand the squares of the binomials:

$(a + b)^2 = a^2 + 2ab + b^2$

$(c + d)^2 = c^2 + 2cd + d^2$

Substituting these expressions back into the previous equation, we get:

$(a + b + c + d)^2 = (a^2 + 2ab + b^2) + 2(a + b)(c + d) + (c^2 + 2cd + d^2)$

Distributing and Combining

Now, we need to distribute the middle term and combine like terms:

$\begin{align*} (a + b + c + d)^2 &= a^2 + 2ab + b^2 + 2ac + 2ad + 2bc + 2bd + c^2 + 2cd + d^2 \
&= \boxed{a^2 + b^2 + c^2 + d^2 + 2ab + 2ac + 2ad + 2bc + 2bd + 2cd} \end{align*}$

Conclusion

In this article, we have successfully expanded the expression (a + b + c + d)^2 using the binomial theorem. The final result is a sum of squared terms and products of pairs of terms. This expansion has numerous applications in mathematics, physics, and engineering, and is an essential tool for problem-solving in these fields.