%5E8-binomial-expansion.jpeg "(1+3x)^8 Binomial Expansion")
Binomial Expansion of (1+3x)^8
In algebra, the binomial theorem is a mathematical formula that describes the expansion of powers of a binomial, which is an expression consisting of two terms. In this article, we will explore the binomial expansion of (1+3x)^8.
What is Binomial Expansion?
Binomial expansion is a process of expanding a binomial expression raised to a power, such as (a+b)^n, into a sum of terms involving various powers of a and b. The binomial theorem provides a formula for expanding such expressions, which can be written as:
$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
where n is a positive integer, and \binom{n}{k} is the binomial coefficient, which can be calculated as:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
Binomial Expansion of (1+3x)^8
Now, let's apply the binomial theorem to expand (1+3x)^8. We can use the formula above, with a=1, b=3x, and n=8.
$(1+3x)^8 = \sum_{k=0}^8 \binom{8}{k} 1^{8-k} (3x)^k$
Let's calculate the binomial coefficients:
$\binom{8}{0} = \frac{8!}{0!8!} = 1$ $\binom{8}{1} = \frac{8!}{1!7!} = 8$ $\binom{8}{2} = \frac{8!}{2!6!} = 28$ $\binom{8}{3} = \frac{8!}{3!5!} = 56$ $\binom{8}{4} = \frac{8!}{4!4!} = 70$ $\binom{8}{5} = \frac{8!}{5!3!} = 56$ $\binom{8}{6} = \frac{8!}{6!2!} = 28$ $\binom{8}{7} = \frac{8!}{7!1!} = 8$ $\binom{8}{8} = \frac{8!}{8!0!} = 1$
Now, let's plug these values into the expansion formula:
$(1+3x)^8 = 1 + 8(3x) + 28(3x)^2 + 56(3x)^3 + 70(3x)^4 + 56(3x)^5 + 28(3x)^6 + 8(3x)^7 + (3x)^8$
Simplifying the expression, we get:
$(1+3x)^8 = 1 + 24x + 336x^2 + 2016x^3 + 7560x^4 + 18144x^5 + 28224x^6 + 28224x^7 + 6561x^8$
And that's the binomial expansion of (1+3x)^8!
Conclusion
In this article, we have explored the binomial expansion of (1+3x)^8 using the binomial theorem. We have seen how to apply the formula to calculate the expansion, and simplified the expression to obtain the final result. Binomial expansion is an important concept in algebra, with many applications in mathematics, physics, and engineering.
