%5E3-binomial-expansion.jpeg "(2x+3)^3 Binomial Expansion")
Binomial Expansion of (2x+3)^3
In algebra, the binomial theorem is a powerful tool for expanding powers of binomials, which are expressions consisting of two terms. One common application of the binomial theorem is to expand expressions of the form $(a+b)^n$, where $a$ and $b$ are constants or variables, and $n$ is a positive integer. In this article, we will focus on expanding the binomial $(2x+3)^3$.
The Binomial Theorem
The binomial theorem states that for any positive integer $n$, the expansion of $(a+b)^n$ is given by:
$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k$
where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.
Expanding (2x+3)^3
To expand $(2x+3)^3$, we can use the binomial theorem with $a = 2x$, $b = 3$, and $n = 3$. We get:
$(2x+3)^3 = \sum_{k=0}^{3} \binom{3}{k} (2x)^{3-k}(3)^k$
Calculating the Coefficients
Now, we need to calculate the coefficients for each term in the expansion. We have:
$\binom{3}{0} = \frac{3!}{0!(3-0)!} = 1$ $\binom{3}{1} = \frac{3!}{1!(3-1)!} = 3$ $\binom{3}{2} = \frac{3!}{2!(3-2)!} = 3$ $\binom{3}{3} = \frac{3!}{3!(3-3)!} = 1$
Expanding the Terms
Now, we can expand each term in the summation:
$(2x)^3(3)^0 = 8x^3$ $(2x)^2(3)^1 = 12x^2$ $(2x)^1(3)^2 = 18x$ $(2x)^0(3)^3 = 27$
Final Expansion
Therefore, the binomial expansion of $(2x+3)^3$ is:
$(2x+3)^3 = \boxed{8x^3 + 36x^2 + 54x + 27}$
This expansion can be used in various mathematical and scientific applications, such as calculus, algebra, and physics.
