F (x) = x³ - 3x² - 4x + 12

Here's a breakdown of the function F(x) = x³ - 3x² - 4x + 12, along with some key things to understand about it: Understanding the Function - Polynomial Function: This is a polynomial function because it's a combination of terms with different powers of 'x'. The highest power of 'x' (in this case, 3) determines the degree of the polynomial.- Cubic Function: Since the highest power of 'x' is 3, this is a cubic function. Key Properties - Roots (or Zeros): The roots of a function are the values of 'x' where the function equals zero (F(x) = 0). Finding the roots is often a key part of analyzing a polynomial function. There are several ways to find the roots (factoring, the Rational Root Theorem, graphing, etc.).- Turning Points: A cubic function has at most two turning points, where the graph changes from increasing to decreasing or vice versa. These turning points are related to the function's derivative.- End Behavior: As 'x' approaches positive infinity, the function will also approach positive infinity. As 'x' approaches negative infinity, the function will approach negative infinity. This is because the leading term (x³) dominates the function's behavior for very large or very small values of 'x'. Example: Finding a Root Let's try to find one root of this function by factoring: 1. Factor by Grouping: Group the terms together:(x³ - 3x²) + (-4x + 12)2. Factor out Common Factors:x²(x - 3) - 4(x - 3)3. Factor out (x - 3):(x - 3)(x² - 4)4. Factor the Difference of Squares:(x - 3)(x + 2)(x - 2)5. Set Each Factor to Zero:x - 3 = 0 => x = 3x + 2 = 0 => x = -2x - 2 = 0 => x = 2 Therefore, x = 3, x = -2, and x = 2 are the roots of this function. Further Exploration - Graphing: Graph the function to visualize its behavior, including the roots and turning points.- Derivative: Calculate the derivative of F(x) to find the turning points and analyze the function's increasing and decreasing intervals.

Answered by aiahbinilat | 2024-10-16