%5E4-expansion.jpeg "(a+b)^4 Expansion")
The Expansion of (a+b)^4
In algebra, the expansion of (a+b)^n is a fundamental concept that is used extensively in various mathematical operations. In this article, we will focus on the expansion of (a+b)^4, which is a specific case of the binomial theorem.
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 = a^n + na^(n-1)b + n(n-1)a^(n-2)b^2 + ... + nab^(n-1) + b^n
where n is a positive integer, and a and b are constants.
Expansion of (a+b)^4
Using the binomial theorem, we can expand (a+b)^4 as follows:
(a+b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4
This expansion can be derived by substituting n = 4 into the binomial theorem formula.
Breakdown of the Expansion
Let's break down the expansion of (a+b)^4 into its individual terms:
- a^4: This is the first term of the expansion, which is obtained by raising a to the power of 4.
- 4a^3b: This term is obtained by multiplying a^3 by b and by the binomial coefficient 4.
- 6a^2b^2: This term is obtained by multiplying a^2 by b^2 and by the binomial coefficient 6.
- 4ab^3: This term is obtained by multiplying a by b^3 and by the binomial coefficient 4.
- b^4: This is the last term of the expansion, which is obtained by raising b to the power of 4.
Conclusion
In conclusion, the expansion of (a+b)^4 is a fundamental concept in algebra that is used extensively in various mathematical operations. By using the binomial theorem, we can derive the expansion of (a+b)^4 as a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4. This expansion has numerous applications in mathematics, physics, engineering, and other fields.
