%5E5-expand.jpeg "(2x+5)^5 Expand")
** Expanded Form of (2x+5)^5 **
In this article, we will explore the expanded form of the expression (2x+5)^5. To do this, we will use the binomial theorem, which provides a formula for expanding powers of a binomial expression.
The Binomial Theorem
The binomial theorem states that for any positive integer n, the following formula holds:
$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.
Expanding (2x+5)^5
Using the binomial theorem, we can expand (2x+5)^5 as follows:
$(2x+5)^5 = \sum_{k=0}^5 \binom{5}{k} (2x)^{5-k} 5^k$
Evaluating the binomial coefficients, we get:
$\binom{5}{0} = 1, \binom{5}{1} = 5, \binom{5}{2} = 10, \binom{5}{3} = 10, \binom{5}{4} = 5, \binom{5}{5} = 1$
Substituting these values into the expansion, we get:
$(2x+5)^5 = (2x)^5 + 5(2x)^4(5) + 10(2x)^3(5)^2 + 10(2x)^2(5)^3 + 5(2x)(5)^4 + (5)^5$
Simplifying each term, we get:
$(2x+5)^5 = 32x^5 + 160x^4 + 400x^3 + 800x^2 + 1250x + 3125$
And that's the expanded form of (2x+5)^5!
