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(x+2)^5: Understanding the Expansion of a Binomial Expression
In algebra, binomial expressions are a fundamental concept, and understanding how to expand them is crucial for problem-solving. One such expression is (x+2)^5, which we will explore in this article.
What is a Binomial Expression?
A binomial expression is a mathematical expression consisting of two terms, typically denoted by a variable (such as x) and a constant. In the case of (x+2)^5, the binomial expression is x+2. The exponent 5 indicates that the expression is raised to the power of 5.
Expanding (x+2)^5 using the Binomial Theorem
The Binomial Theorem is a mathematical formula that enables us to expand a binomial expression raised to a power. The formula is:
(a+b)^n = a^n + na^(n-1)b + n(n-1)a^(n-2)b^2 + … + b^n
In our case, a = x, b = 2, and n = 5. Plugging these values into the formula, we get:
(x+2)^5 = x^5 + 5x^4(2) + 10x^3(2)^2 + 10x^2(2)^3 + 5x(2)^4 + (2)^5
Simplifying the Expanded Expression
Let's simplify the expanded expression by evaluating the powers of 2 and combining like terms:
(x+2)^5 = x^5 + 10x^4 + 40x^3 + 80x^2 + 80x + 32
Properties of (x+2)^5
Understanding the properties of (x+2)^5 can help us solve problems and identify patterns:
- Symmetry: The expression
(x+2)^5is symmetric about the vertical axisx = -2. - Degree: The degree of the expression is
5, indicating that the highest power ofxis5. - Coefficients: The coefficients of the terms in the expanded expression follow a specific pattern, which can be useful in problem-solving.
Conclusion
In this article, we explored the expansion of the binomial expression (x+2)^5 using the Binomial Theorem. We simplified the expanded expression and discussed some of its properties. Understanding how to expand and work with binomial expressions is essential in algebra and has numerous applications in mathematics, science, and engineering.
