(2x+3)^5 Binomial Theorem

4 min read Jul 03, 2024

Binomial Theorem: Expanding (2x+3)^5

Introduction

The binomial theorem is a powerful tool in algebra that allows us to expand powers of binomials, such as (2x+3)^5. In this article, we will explore the binomial theorem and use it to expand (2x+3)^5.

What is the Binomial Theorem?

The binomial theorem is a mathematical formula that describes the expansion of powers of a binomial, which is an expression consisting of two terms. The theorem states that:

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

where n is a positive integer, a and b are constants, and \binom{n}{k} is the binomial coefficient.

Expanding (2x+3)^5

To expand (2x+3)^5, we can use the binomial theorem. We will let a = 2x and b = 3, and n = 5. Then, we can write:

$(2x+3)^5 = \sum_{k=0}^{5} \binom{5}{k} (2x)^{5-k} 3^k$

Calculating the Binomial Coefficients

To calculate the binomial coefficients, we can use the formula:

$\binom{n}{k} = \frac{n!}{k! (n-k)!}$

where n! is the factorial of n. In our case, we need to calculate the binomial coefficients for k = 0, 1, 2, 3, 4, and 5. These are:

\binom{5}{0} = 1
\binom{5}{1} = 5
\binom{5}{2} = 10
\binom{5}{3} = 10
\binom{5}{4} = 5
\binom{5}{5} = 1

Expanding the Expression

Now, we can expand the expression using the binomial theorem and the binomial coefficients we calculated:

$(2x+3)^5 = \binom{5}{0} (2x)^5 3^0 + \binom{5}{1} (2x)^4 3^1 + \binom{5}{2} (2x)^3 3^2 + \binom{5}{3} (2x)^2 3^3 + \binom{5}{4} (2x)^1 3^4 + \binom{5}{5} (2x)^0 3^5$

Simplifying the expression, we get:

$(2x+3)^5 = 32x^5 + 240x^4 * 3 + 480x^3 * 3^2 + 240x^2 * 3^3 + 60x * 3^4 + 3^5$

Simplifying further, we get:

$(2x+3)^5 = 32x^5 + 720x^4 + 4320x^3 + 6480x^2 + 4860x + 243$

Conclusion

In this article, we used the binomial theorem to expand (2x+3)^5. We calculated the binomial coefficients and expanded the expression using the theorem. The final result is a polynomial expression in x. The binomial theorem is a powerful tool in algebra, and it has many applications in mathematics, physics, and other fields.