%5E3-expand.jpeg "(x-2)^3 Expand")
Expanding (x-2)^3
In mathematics, expanding an expression means rewriting it in a simpler form, without any exponents. This is often done to make the expression easier to work with, or to understand its behavior. Today, we'll explore how to expand the expression (x-2)^3.
Understanding the Concept
The expression (x-2)^3 means we are multiplying (x-2) by itself three times: (x-2) * (x-2) * (x-2)
Methods of Expansion
There are two main ways to expand (x-2)^3:
1. Direct Multiplication:
- Step 1: Multiply the first two factors: (x-2) * (x-2) = x^2 - 4x + 4
- Step 2: Multiply the result from step 1 by the remaining factor (x-2): (x^2 - 4x + 4) * (x-2) = x^3 - 6x^2 + 12x - 8
2. Binomial Theorem:
The Binomial Theorem provides a general formula for expanding any expression of the form (a+b)^n. In our case, a = x, b = -2, and n = 3.
The Binomial Theorem states: (a+b)^n = a^n + na^(n-1)b + (n(n-1)/2!)a^(n-2)b^2 + ... + b^n
Applying this to our expression: (x-2)^3 = x^3 + 3x^2(-2) + 3x(-2)^2 + (-2)^3 = x^3 - 6x^2 + 12x - 8
The Result
No matter which method you use, the expanded form of (x-2)^3 is: x^3 - 6x^2 + 12x - 8
Understanding the Expansion
The expanded form provides valuable insights:
- Degree: The highest power of x is 3, indicating a cubic polynomial.
- Coefficients: The coefficients -6, 12, and -8 represent the constants multiplying the different powers of x.
- Constant Term: The constant term -8 is obtained by multiplying (-2) three times.
Conclusion
Expanding expressions like (x-2)^3 is a fundamental skill in algebra. Understanding the process and its implications allows you to manipulate and interpret expressions effectively.