(x%2B4).jpeg "(x-1)(x+4)")
Factoring and Solving the Expression (x-1)(x+4)
The expression (x-1)(x+4) represents a product of two binomials. Understanding how to factor and solve this expression is crucial in algebra and other mathematical fields. Let's explore its properties and applications:
Factoring the Expression
The expression is already factored. It is the product of two linear factors: (x-1) and (x+4). However, we can also expand the expression to get a quadratic expression:
(x-1)(x+4) = x(x+4) - 1(x+4) = x² + 4x - x - 4 = x² + 3x - 4
Finding the Roots of the Expression
The roots of the expression are the values of x that make the expression equal to zero. Since the expression is factored, we can easily find the roots by setting each factor to zero:
- x - 1 = 0 --> x = 1
- x + 4 = 0 --> x = -4
Therefore, the roots of the expression (x-1)(x+4) are x = 1 and x = -4.
Graphing the Expression
The expression (x-1)(x+4) represents a parabola when graphed. The roots of the expression, x = 1 and x = -4, are the x-intercepts of the parabola. The parabola opens upwards because the coefficient of the x² term is positive.
Applications
This expression and its properties are used in various contexts, including:
- Solving Quadratic Equations: The factored form helps find the solutions to quadratic equations of the form x² + 3x - 4 = 0.
- Finding the Zeros of a Function: The roots of the expression represent the zeros of the function f(x) = (x-1)(x+4).
- Analyzing the Behavior of a Quadratic Function: The factored form provides information about the intercepts, vertex, and overall shape of the graph.
In conclusion, understanding the factored form (x-1)(x+4) and its properties is essential for solving various mathematical problems and analyzing the behavior of quadratic expressions and functions.