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(a+1)^4: Understanding the Binomial Expansion
In algebra, one of the most powerful tools for expanding expressions involving binomials is the binomial theorem. In this article, we will explore the expansion of (a+1)^4, a classic example of a binomial expression.
The Binomial Theorem
The binomial theorem is a mathematical formula that describes the expansion of powers of a binomial expression. It is stated as follows:
(a+b)^n = a^n + na^(n-1)b + n(n-1)a^(n-2)b^2 + ... + b^n
where a and b are constants, and n is a positive integer.
Expanding (a+1)^4
Using the binomial theorem, we can expand (a+1)^4 as follows:
(a+1)^4 = a^4 + 4a^3(1) + 6a^2(1)^2 + 4a(1)^3 + (1)^4
Simplifying the expression, we get:
(a+1)^4 = a^4 + 4a^3 + 6a^2 + 4a + 1
Breaking Down the Expansion
Let's break down the expansion to understand each term:
a^4: This is the first term, which is simplyaraised to the power of 4.4a^3: This is the second term, which is 4 timesaraised to the power of 3.6a^2: This is the third term, which is 6 timesaraised to the power of 2.4a: This is the fourth term, which is 4 timesa.1: This is the fifth and final term, which is simply 1.
Conclusion
In this article, we have explored the expansion of (a+1)^4 using the binomial theorem. By applying the theorem, we obtained the expanded expression a^4 + 4a^3 + 6a^2 + 4a + 1. This expansion is a fundamental concept in algebra and is used extensively in various mathematical and scientific applications.
