(x-1)(x%2B3)(x%2B4).jpeg "(x-2)(x-1)(x+3)(x+4)")
Expanding and Factoring (x-2)(x-1)(x+3)(x+4)
This article explores the process of expanding and factoring the expression (x-2)(x-1)(x+3)(x+4).
Expanding the Expression
To expand the expression, we can use the distributive property, also known as FOIL (First, Outer, Inner, Last), repeatedly.
- First: Multiply the first terms of the first two binomials: (x-2)(x-1) = x² - 3x + 2
- Second: Multiply the first two binomials and the third binomial: (x² - 3x + 2)(x+3) = x³ - x² - 7x + 6
- Third: Multiply the resulting trinomial with the last binomial: (x³ - x² - 7x + 6)(x+4) = x⁴ + 3x³ - 21x² - 10x + 24
Therefore, the expanded form of the expression is x⁴ + 3x³ - 21x² - 10x + 24.
Factoring the Expression
The process of factoring this expression involves reversing the expansion steps. However, since the expression is already factored, the simplest factored form is: (x-2)(x-1)(x+3)(x+4).
This form highlights the roots of the polynomial, which are the values of x that make the expression equal to zero. The roots are x = 2, x = 1, x = -3, and x = -4.
Applications
Expanding and factoring expressions like this have various applications in mathematics and related fields. Some examples include:
- Solving polynomial equations: Finding the roots of a polynomial equation is essential for solving many problems in algebra, calculus, and other areas.
- Graphing functions: The factored form of a polynomial can be used to easily determine the x-intercepts of its graph.
- Calculus: Understanding the expansion and factoring of polynomials is essential for calculating derivatives and integrals.
- Engineering and physics: Polynomial expressions are used to model various physical phenomena, such as the trajectory of a projectile or the behavior of circuits.
Conclusion
Understanding the process of expanding and factoring polynomials is crucial for various mathematical applications. The expression (x-2)(x-1)(x+3)(x+4) serves as a simple example to demonstrate these concepts. By mastering these techniques, you can efficiently solve complex problems involving polynomials and their related applications.