%5E8-binomial-expansion.jpeg "(x+3)^8 Binomial Expansion")
Binomial Expansion of (x+3)^8
In algebra, the binomial theorem is a powerful tool for expanding powers of binomials, which are expressions consisting of two terms. In this article, we will explore the binomial expansion of (x+3)^8.
The Binomial Theorem
The binomial theorem states that for any positive integer n, the following equation holds:
$(x+y)^n = \sum_{k=0}^n {n \choose k} x^{n-k} y^k$
where {n \choose k} is the binomial coefficient, which can be calculated as:
${n \choose k} = \frac{n!}{k!(n-k)!}$
Expanding (x+3)^8
Using the binomial theorem, we can expand (x+3)^8 as follows:
$(x+3)^8 = \sum_{k=0}^8 {8 \choose k} x^{8-k} 3^k$
To calculate the binomial coefficients, we can use the formula above:
${8 \choose 0} = 1$ ${8 \choose 1} = 8$ ${8 \choose 2} = 28$ ${8 \choose 3} = 56$ ${8 \choose 4} = 70$ ${8 \choose 5} = 56$ ${8 \choose 6} = 28$ ${8 \choose 7} = 8$ ${8 \choose 8} = 1$
Now, we can write out the expansion:
$(x+3)^8 = x^8 + 8x^7(3) + 28x^6(3^2) + 56x^5(3^3) + 70x^4(3^4) + 56x^3(3^5) + 28x^2(3^6) + 8x(3^7) + 3^8$
Simplifying the expression, we get:
$(x+3)^8 = x^8 + 24x^7 + 252x^6 + 1512x^5 + 5670x^4 + 13608x^3 + 20592x^2 + 17424x + 6561$
And that's the binomial expansion of (x+3)^8!
Conclusion
In this article, we have explored the binomial expansion of (x+3)^8 using the binomial theorem. We have calculated the binomial coefficients and written out the expansion. The final result is a polynomial expression with terms of increasing degree in x. This expansion can be useful in various algebraic manipulations and applications.
