Finding Zeros Of Polynomial Functions Solving H(x) = (x^2 - 49)(x + 4)

JU07/23/2025 08, 2025 by THE IDEN 71 views

Unveiling the Zeros of a Polynomial Function: A Comprehensive Guide to Solving h(x) = (x^2 - 49)(x + 4)

Zeros of a polynomial function are the values of x that make the function equal to zero. In simpler terms, they are the points where the graph of the function intersects the x-axis. Finding these zeros is a fundamental concept in algebra and calculus, as they provide critical information about the behavior of the function. In this article, we delve into the process of finding the zeros of a given polynomial function, h(x) = (x^2 - 49)(x + 4), providing a step-by-step guide and a comprehensive explanation of the underlying principles.

Defining Polynomial Functions and Their Significance

Polynomial functions are a cornerstone of mathematics, characterized by their smooth, continuous curves and their ability to model a wide range of real-world phenomena. They are defined as expressions involving variables raised to non-negative integer powers, combined with coefficients and constants. The degree of a polynomial is the highest power of the variable in the expression, and it dictates the overall shape and behavior of the function. For instance, a linear function (degree 1) is a straight line, a quadratic function (degree 2) is a parabola, and so on.

Polynomial functions are ubiquitous in various fields, including physics, engineering, economics, and computer science. They are used to model projectile motion, electrical circuits, population growth, and many other phenomena. Understanding their properties, including finding their zeros, is crucial for solving problems and making predictions in these fields.

The Significance of Zeros

Zeros of a polynomial function, also known as roots, hold significant information about the function's behavior. They are the points where the function's value is zero, indicating where the graph intersects the x-axis. These points are critical for understanding the function's overall shape, its intervals of increase and decrease, and its local maxima and minima.

Zeros also play a crucial role in solving polynomial equations. Finding the zeros of a polynomial is equivalent to solving the equation formed by setting the polynomial equal to zero. This is a fundamental problem in algebra, with applications in various fields. For example, in physics, finding the zeros of a polynomial can help determine the time at which a projectile hits the ground, or the equilibrium points of a system.

Factoring and the Zero Product Property

One of the key techniques for finding zeros is factoring. Factoring involves expressing a polynomial as a product of simpler expressions, typically linear or quadratic factors. The zero product property states that if the product of two or more factors is zero, then at least one of the factors must be zero. This property is the foundation for finding zeros from factored polynomials. If we can factor a polynomial, we can set each factor equal to zero and solve for x, thereby finding the zeros of the polynomial.

Step 1: Recognize the Factored Form

The given function, h(x) = (x^2 - 49)(x + 4), is already presented in a partially factored form. This is a significant advantage, as it simplifies the process of finding the zeros. We can observe that the first factor, (x^2 - 49), is a difference of squares, a common pattern that can be easily factored further. The second factor, (x + 4), is a linear term and cannot be factored further.

Step 2: Factor the Difference of Squares

The difference of squares pattern states that a^2 - b^2 = (a + b)(a - b). Applying this pattern to the first factor, (x^2 - 49), we can identify that x^2 is the square of x and 49 is the square of 7. Therefore, we can factor (x^2 - 49) as (x + 7)(x - 7). Now, the function h(x) can be written in fully factored form as h(x) = (x + 7)(x - 7)(x + 4).

Step 3: Apply the Zero Product Property

The zero product property states that if the product of several factors is zero, then at least one of the factors must be zero. In our case, the product of the three factors (x + 7), (x - 7), and (x + 4) is equal to h(x), which we want to be zero. Therefore, we can set each factor equal to zero and solve for x:

x + 7 = 0
x - 7 = 0
x + 4 = 0

Step 4: Solve for x

Solving each of the equations above, we get:

x = -7
x = 7
x = -4

These are the zeros of the function h(x). They are the x-values that make the function equal to zero. In other words, the graph of h(x) intersects the x-axis at the points (-7, 0), (7, 0), and (-4, 0).

Among the given options (A. 0, B. 4, C. 7, D. 49), only 7 is a zero of the function h(x). Therefore, the correct answer is C. 7. This means that when x = 7, the function h(x) evaluates to zero.

Verifying the Solution

To ensure our solution is correct, we can substitute each of the zeros we found back into the original function h(x) and verify that the result is indeed zero. For example, let's substitute x = 7 into h(x):

h(7) = (7^2 - 49)(7 + 4) = (49 - 49)(11) = (0)(11) = 0

Similarly, we can verify that h(-7) = 0 and h(-4) = 0. This confirms that our solutions are correct.

The zeros of a polynomial function have a clear graphical interpretation. They are the x-coordinates of the points where the graph of the function intersects the x-axis. These points are also known as x-intercepts. The graph of h(x) = (x^2 - 49)(x + 4) would intersect the x-axis at x = -7, x = 7, and x = -4.

A graph can provide a visual confirmation of the zeros we found algebraically. By plotting the function, we can visually identify the x-intercepts and verify that they correspond to the zeros we calculated. The graph also provides insights into the function's overall behavior, such as its intervals of increase and decrease, and its local maxima and minima.

Finding the zeros of a polynomial function is a fundamental concept in algebra and calculus. It involves identifying the values of x that make the function equal to zero. The process typically involves factoring the polynomial and applying the zero product property. The zeros of a function provide critical information about its behavior, its graph, and its applications in various fields.

In this article, we have provided a comprehensive guide to finding the zeros of the polynomial function h(x) = (x^2 - 49)(x + 4). We have demonstrated the steps involved in factoring the function, applying the zero product property, and solving for x. We have also emphasized the graphical interpretation of zeros and their significance in understanding the behavior of polynomial functions. By mastering these concepts, you will be well-equipped to tackle more complex problems involving polynomial functions and their applications.