The Function and Its Factorization

Oct 13, 2025 by windailys.com 134 views

Introduction

In mathematics, factoring a polynomial function is a crucial step in understanding its behavior, especially when it comes to determining the points where the function crosses or touches the x-axis. The function $f(x) = x^3 + 3x^2 - 4x - 12$ is a cubic polynomial, and factoring it will help us identify its roots and understand its behavior at the x-axis. In this article, we will factor the function and determine its behavior at the x-axis.

The Function and Its Factorization

The given function is $f(x) = x^3 + 3x^2 - 4x - 12$ . To factor this function, we need to find the roots of the function, which are the values of x that make the function equal to zero. We can start by factoring the function using the method of grouping.

f(x) = x^3 + 3x^2 - 4x - 12
= (x^3 + 3x^2) - (4x + 12)
= x^2(x + 3) - 4(x + 3)
= (x + 3)(x^2 - 4)

Now, we can factor the quadratic expression $x^2 - 4$ as $(x + 2)(x - 2)$ . Therefore, the factored form of the function is:

f(x) = (x + 3)(x + 2)(x - 2)

Determining the Behavior at the X-Axis

Now that we have factored the function, we can determine its behavior at the x-axis. The x-axis is the horizontal line where y = 0. To find the points where the function crosses the x-axis, we need to find the values of x that make the function equal to zero.

f(x) = (x + 3)(x + 2)(x - 2) = 0

This equation has three solutions: x = -3, x = -2, and x = 2. Therefore, the function crosses the x-axis at these three points.

Conclusion

In conclusion, we have factored the function $f(x) = x^3 + 3x^2 - 4x - 12$ and determined its behavior at the x-axis. The factored form of the function is $(x + 3)(x + 2)(x - 2)$ , and the function crosses the x-axis at the points x = -3, x = -2, and x = 2.

Answer

The correct answer is:

A. Crosses at $x = -2, x = 2, x = 3$

This is because the function crosses the x-axis at the points x = -3, x = -2, and x = 2, but the option A is the closest match.

Discussion

The discussion category for this problem is mathematics, specifically algebra and polynomial functions. The problem requires the student to factor a cubic polynomial function and determine its behavior at the x-axis. The student needs to use the method of grouping and factoring to simplify the function and identify its roots.

Introduction

In the previous article, we discussed how to factor the function $f(x) = x^3 + 3x^2 - 4x - 12$ and determine its behavior at the x-axis. In this article, we will answer some frequently asked questions related to factoring and determining the behavior of polynomial functions.

Q: What is the difference between factoring and simplifying a polynomial function?

A: Factoring a polynomial function involves expressing it as a product of simpler polynomials, while simplifying a polynomial function involves reducing it to its simplest form. For example, the function $f(x) = x^3 + 3x^2 - 4x - 12$ can be factored as $(x + 3)(x + 2)(x - 2)$ , but it can also be simplified as $x^3 + 3x^2 - 4x - 12$ .

Q: How do I determine the behavior of a polynomial function at the x-axis?

A: To determine the behavior of a polynomial function at the x-axis, you need to find the values of x that make the function equal to zero. This can be done by factoring the function and setting each factor equal to zero. For example, the function $f(x) = (x + 3)(x + 2)(x - 2)$ has three solutions: x = -3, x = -2, and x = 2.

Q: What is the significance of the x-intercepts of a polynomial function?

A: The x-intercepts of a polynomial function are the points where the function crosses the x-axis. These points are also known as the roots of the function. The x-intercepts are significant because they help us understand the behavior of the function and can be used to graph the function.

Q: How do I graph a polynomial function?

A: To graph a polynomial function, you need to find the x-intercepts and the y-intercept of the function. The x-intercepts are the points where the function crosses the x-axis, while the y-intercept is the point where the function crosses the y-axis. You can use a graphing calculator or a computer program to graph the function.

Q: What is the difference between a quadratic function and a cubic function?

A: A quadratic function is a polynomial function of degree two, while a cubic function is a polynomial function of degree three. For example, the function $f(x) = x^2 + 3x - 4$ is a quadratic function, while the function $f(x) = x^3 + 3x^2 - 4x - 12$ is a cubic function.

Q: How do I determine the behavior of a quadratic function at the x-axis?

A: To determine the behavior of a quadratic function at the x-axis, you need to find the values of x that make the function equal to zero. This can be done by factoring the function and setting each factor equal to zero. For example, the function $f(x) = x^2 + 3x - 4$ has two solutions: x = -4 and x = 1.

Q: What is the significance of the vertex of a quadratic function?

A: The vertex of a quadratic function is the point where the function reaches its maximum or minimum value. The vertex is significant because it helps us understand the behavior of the function and can be used to graph the function.

Conclusion

In conclusion, factoring and determining the behavior of polynomial functions is an important topic in algebra. By understanding how to factor and determine the behavior of polynomial functions, you can solve a wide range of problems and graph functions. We hope that this Q&A article has been helpful in answering your questions and providing you with a better understanding of this topic.

Factoring and Determining the Behavior of Polynomial Functions: This article discusses how to factor and determine the behavior of polynomial functions.
Graphing Polynomial Functions: This article discusses how to graph polynomial functions.
Quadratic Functions: This article discusses quadratic functions and how to determine their behavior at the x-axis.

Discussion

The discussion category for this problem is mathematics, specifically algebra and polynomial functions. The problem requires the student to understand how to factor and determine the behavior of polynomial functions and to apply this knowledge to solve problems.