%5E3-expand.jpeg "(2x+5)^3 Expand")
Expanding (2x+5)^3
When we encounter an expression like (2x+5)^3, we need to expand it to simplify it. Expanding an expression means to multiply it out and combine like terms. In this case, we will use the binomial theorem to expand the cube of the binomial 2x+5.
The Binomial Theorem
The binomial theorem is a formula for expanding powers of a binomial, which is an expression consisting of two terms. The theorem states that:
$(a+b)^n = \sum_{k=0}^n {n \choose k} a^{n-k}b^k$
where ${n \choose k}$ is the binomial coefficient, defined as:
${n \choose k} = \frac{n!}{k!(n-k)!}$
Expanding (2x+5)^3
Using the binomial theorem, we can expand (2x+5)^3 as follows:
$(2x+5)^3 = (2x)^3 + 3(2x)^2(5) + 3(2x)(5)^2 + (5)^3$
Simplifying each term, we get:
$(2x)^3 = 8x^3$
$3(2x)^2(5) = 3(4x^2)(5) = 60x^2$
$3(2x)(5)^2 = 3(2x)(25) = 150x$
$(5)^3 = 125$
Now, we can combine these terms to get the final expansion:
$(2x+5)^3 = 8x^3 + 60x^2 + 150x + 125$
And that's the result!
Conclusion
In this article, we learned how to expand (2x+5)^3 using the binomial theorem. We saw how to apply the theorem to simplify the expression and get the final result. Expanding binomials is an important skill in algebra, and with practice, you'll become proficient in no time.
