%5E5-expand.jpeg "(a+b)^5 Expand")
(a+b)^5 Expansion
In algebra, the expansion of (a+b)^5 is an important concept in understanding binomial theorem. In this article, we will discuss the expansion of (a+b)^5 and its formula.
What is Binomial Theorem?
The binomial theorem is a mathematical formula that describes the expansion of powers of a binomial, which is an algebraic expression consisting of two terms. The binomial theorem is written as:
(a+b)^n = ∑[n choose k] * a^(n-k) * b^k
where n is the power, k is the index, and [n choose k] is the binomial coefficient.
Expansion of (a+b)^5
To expand (a+b)^5, we can use the binomial theorem with n=5. The expansion is given by:
(a+b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5
This expansion can be calculated by using the formula above and substituting n=5. The result is a polynomial expression with six terms, each with a different power of a and b.
Explanation of Each Term
Let's break down each term in the expansion:
a^5
This term is the result of multiplying a by itself five times, since there is no b present.
5a^4b
This term is the result of multiplying a by itself four times and b once, with a coefficient of 5.
10a^3b^2
This term is the result of multiplying a by itself three times and b twice, with a coefficient of 10.
10a^2b^3
This term is the result of multiplying a by itself twice and b three times, with a coefficient of 10.
5ab^4
This term is the result of multiplying a once and b four times, with a coefficient of 5.
b^5
This term is the result of multiplying b by itself five times, since there is no a present.
Conclusion
In conclusion, the expansion of (a+b)^5 is a polynomial expression with six terms, each with a different power of a and b. This expansion is an important concept in algebra and is used in various mathematical applications.
