(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.