(x-1)(x-3).jpeg "(x+2)(x-1)(x-3)")
Factoring and Expanding the Expression (x+2)(x-1)(x-3)
This expression represents the product of three binomial factors: (x+2), (x-1), and (x-3). We can analyze this expression in two ways: by expanding it or by understanding its factored form.
Expanding the Expression
To expand the expression, we can use the distributive property (or FOIL method) multiple times.
Step 1: Expand the first two factors.
(x+2)(x-1) = x(x-1) + 2(x-1) = x² - x + 2x - 2 = x² + x - 2
Step 2: Multiply the result by the remaining factor.
(x² + x - 2)(x-3) = x²(x-3) + x(x-3) - 2(x-3) = x³ - 3x² + x² - 3x - 2x + 6 = x³ - 2x² - 5x + 6
Therefore, the expanded form of the expression is x³ - 2x² - 5x + 6.
Understanding the Factored Form
The factored form (x+2)(x-1)(x-3) reveals crucial information about the expression:
- Roots: The factored form directly shows the roots of the expression, which are the values of x that make the expression equal to zero. In this case, the roots are x = -2, x = 1, and x = 3. This is because each factor becomes zero when x equals its respective root.
- X-intercepts: The roots also represent the x-intercepts of the function defined by this expression. These are the points where the graph of the function crosses the x-axis.
Applications
Understanding factored forms and expansions is important in various mathematical contexts, including:
- Solving equations: By setting the expanded form of the expression equal to zero, we can solve for x and find the roots.
- Graphing functions: Knowing the roots helps us to sketch the graph of the function represented by the expression.
- Calculus: Factored forms are useful for finding critical points, inflection points, and other properties of functions.
In summary, (x+2)(x-1)(x-3) represents a polynomial expression that can be expanded or analyzed in its factored form. Both representations provide valuable insights into the behavior of the expression and its corresponding function.