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.