5-binomial-expansion.jpeg "(1-2 X)5 Binomial Expansion")
Binomial Expansion: Understanding (1-2x)^5
Binomial expansion is a fundamental concept in algebra, allowing us to simplify complex expressions by breaking them down into more manageable terms. One of the most common forms of binomial expansion is the expression (a+b)^n, where a and b are variables and n is a positive integer. In this article, we will focus on the specific case of (1-2x)^5, exploring its expansion and providing examples to illustrate the process.
The Binomial Theorem
Before diving into the expansion of (1-2x)^5, it's essential to understand the binomial theorem. This theorem states that for any positive integer n:
$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$
where a and b are variables, and n is a positive integer. The binomial coefficient binom{n}{k} is defined as:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
Expanding (1-2x)^5
Using the binomial theorem, we can expand (1-2x)^5 as follows:
$(1-2x)^5 = \sum_{k=0}^{5} \binom{5}{k} 1^{5-k} (-2x)^k$
Simplifying the expression, we get:
$(1-2x)^5 = \binom{5}{0} 1^5 + \binom{5}{1} 1^4 (-2x)^1 + \binom{5}{2} 1^3 (-2x)^2 + \binom{5}{3} 1^2 (-2x)^3 + \binom{5}{4} 1^1 (-2x)^4 + \binom{5}{5} 1^0 (-2x)^5$
Now, let's evaluate each term:
$\binom{5}{0} 1^5 = 1$
$\binom{5}{1} 1^4 (-2x)^1 = 5(-2x) = -10x$
$\binom{5}{2} 1^3 (-2x)^2 = 10(-2x)^2 = 40x^2$
$\binom{5}{3} 1^2 (-2x)^3 = 10(-2x)^3 = -80x^3$
$\binom{5}{4} 1^1 (-2x)^4 = 5(-2x)^4 = 80x^4$
$\binom{5}{5} 1^0 (-2x)^5 = (-2x)^5 = -32x^5$
Final Expansion
Combining the terms, we arrive at the final expansion of (1-2x)^5:
$(1-2x)^5 = 1 - 10x + 40x^2 - 80x^3 + 80x^4 - 32x^5$
Conclusion
In this article, we have explored the binomial expansion of (1-2x)^5, demonstrating how to apply the binomial theorem to simplify complex expressions. By following the step-by-step process, you can expand any binomial expression, unraveling the mysteries of algebra.
