%5E3-binomial-theorem.jpeg "(x+3)^3 Binomial Theorem")
Binomial Theorem: Expanding (x+3)^3
In algebra, the binomial theorem is a powerful tool for expanding powers of a binomial expression, which is an expression consisting of two terms. One of the most common applications of the binomial theorem is to expand expressions of the form (x+a)^n, where n is a positive integer. In this article, we will focus on expanding the expression (x+3)^3 using the binomial theorem.
The Binomial Theorem Formula
The binomial theorem formula is given by:
(x+a)^n = x^n + nx^(n-1)a + n(n-1)x^(n-2)a^2/2! + … + a^n
where n is a positive integer, and x and a are variables.
Expanding (x+3)^3
To expand (x+3)^3, we can use the binomial theorem formula by substituting x+a = x+3 and n = 3. We get:
(x+3)^3 = x^3 + 3x^2(3) + 3(2)x(3)^2/2! + 3^3
Simplifying the expression, we get:
(x+3)^3 = x^3 + 9x^2 + 27x + 27
Breakdown of the Expansion
Let's break down the expansion to understand the pattern:
- x^3: This is the first term, which is x raised to the power of 3.
- 9x^2: This is the second term, which is 3 times x squared times 3.
- 27x: This is the third term, which is 3 times 2 times x times 3 squared divided by 2 factorial (2!).
- 27: This is the last term, which is 3 cubed.
Conclusion
In this article, we have used the binomial theorem to expand the expression (x+3)^3. The expanded form of the expression is x^3 + 9x^2 + 27x + 27. This expansion can be useful in various algebraic manipulations and applications in mathematics and physics.
