Finding Zeros Of The Polynomial Function P(x) = (x^2 + 4x + 3)(x^2 - 4)

In mathematics, determining the zeros of a polynomial function is a fundamental task with significant applications across various fields. The zeros, also known as roots or x-intercepts, are the values of x for which the polynomial p(x) equals zero. Understanding how to find these zeros is crucial for analyzing the behavior of polynomial functions and solving related problems.

This article delves into the process of finding the zeros of the polynomial function:

$p(x) = (x^2 + 4x + 3)(x^2 - 4)$

We will explore the techniques involved in factoring the polynomial and identifying its roots, providing a comprehensive guide for students and enthusiasts alike.

Factoring the Polynomial

To find the zeros of the polynomial $p(x) = (x^2 + 4x + 3)(x^2 - 4)$, the first step is to factor the polynomial completely. This involves breaking down each quadratic expression into simpler linear factors. By factoring, we transform the polynomial into a product of linear terms, making it easier to identify the values of x that make the polynomial equal to zero. This process not only simplifies the equation but also reveals the underlying structure of the polynomial function, providing insights into its behavior and properties. Factoring is a crucial technique in algebra, with applications ranging from solving equations to simplifying complex expressions.

Factoring the First Quadratic

The first quadratic expression is $x^2 + 4x + 3$. We look for two numbers that multiply to 3 and add to 4. These numbers are 3 and 1. Therefore, we can factor the quadratic as:

$x^2 + 4x + 3 = (x + 3)(x + 1)$

This factorization breaks down the quadratic expression into two linear factors, each representing a root of the quadratic. The numbers 3 and 1 are key to this factorization, as they satisfy the necessary conditions for the coefficients of the x terms and the constant term. Understanding how to find these numbers is crucial for mastering quadratic factorization. This skill is not only useful for solving polynomial equations but also for simplifying rational expressions and analyzing graphs of quadratic functions.

Factoring the Second Quadratic

The second quadratic expression is $x^2 - 4$. This is a difference of squares, which can be factored as:

$x^2 - 4 = (x - 2)(x + 2)$

The difference of squares factorization is a common pattern in algebra, where the difference between two perfect squares can be factored into the product of the sum and difference of their square roots. In this case, $x^2$ and 4 are both perfect squares, with square roots x and 2, respectively. Recognizing this pattern allows for quick and efficient factorization. This technique is widely applicable in various mathematical contexts, including calculus and complex number theory, making it an essential tool for any mathematician or student.

Complete Factorization

Combining the factored forms of both quadratic expressions, we get the complete factorization of the polynomial:

$p(x) = (x + 3)(x + 1)(x - 2)(x + 2)$

This complete factorization represents the polynomial as a product of four linear factors, each corresponding to a root of the polynomial. The factored form makes it straightforward to identify the zeros of the polynomial, as each factor can be set to zero to find the corresponding root. The complete factorization not only simplifies the process of finding roots but also provides a clear representation of the polynomial's structure, revealing its degree and leading coefficient. This comprehensive understanding of the polynomial's factored form is invaluable for analyzing its behavior and properties.

Identifying the Zeros

Once the polynomial is fully factored, finding the zeros involves setting each factor equal to zero and solving for x. This method is based on the zero-product property, which states that if the product of several factors is zero, then at least one of the factors must be zero. By applying this property, we can systematically find all the values of x that make the polynomial equal to zero. This process transforms the problem of finding zeros into a series of simpler equations, each of which can be solved independently. Understanding the zero-product property is essential for solving polynomial equations and for grasping the relationship between the roots and factors of a polynomial.

Setting Each Factor to Zero

We set each factor of the polynomial to zero:

• $x + 3 = 0$
• $x + 1 = 0$
• $x - 2 = 0$
• $x + 2 = 0$

Each of these equations represents a linear equation that can be easily solved for x. The process of setting each factor to zero transforms the original polynomial equation into a set of simpler equations, making the problem more manageable. This step is crucial for applying the zero-product property effectively and for identifying all the roots of the polynomial. By solving each linear equation, we obtain the values of x that make the corresponding factor equal to zero, thereby revealing the zeros of the polynomial.

Solving for x

Solving each equation for x gives us the zeros of the polynomial:

• From $x + 3 = 0$, we get $x = -3$.
• From $x + 1 = 0$, we get $x = -1$.
• From $x - 2 = 0$, we get $x = 2$.
• From $x + 2 = 0$, we get $x = -2$.

These values of x are the zeros of the polynomial, representing the points where the graph of the polynomial intersects the x-axis. Each zero corresponds to a root of the polynomial equation and provides valuable information about the polynomial's behavior. The zeros are crucial for analyzing the polynomial's graph, determining its intervals of positivity and negativity, and understanding its end behavior. By finding these zeros, we gain a comprehensive understanding of the polynomial function and its properties.

The Zeros

Therefore, the zeros of the polynomial $p(x) = (x^2 + 4x + 3)(x^2 - 4)$ are:

• $x = -3$
• $x = -1$
• $x = 2$
• $x = -2$

These are the x-intercepts of the graph of the polynomial. At these points, the polynomial's value is zero, indicating where the graph crosses or touches the x-axis. The zeros provide essential information about the polynomial's behavior and are fundamental for graphing and analyzing polynomial functions. Each zero corresponds to a factor of the polynomial, and together, they completely determine the polynomial's roots. Understanding these zeros allows for a deeper understanding of the polynomial's structure and its relationship to the x-axis.

Graphing the Polynomial

Plotting the zeros on a graph provides a visual representation of the polynomial's behavior. The zeros are the points where the graph intersects the x-axis, and their location helps to sketch the overall shape of the polynomial. By plotting these points, we can begin to visualize the polynomial's curve, its turning points, and its end behavior. Graphing the polynomial not only confirms the accuracy of the calculated zeros but also provides a visual understanding of the polynomial's properties and characteristics.

Plotting the Zeros

We plot the zeros $x = -3$, $x = -1$, $x = 2$, and $x = -2$ on the x-axis.

These points are the x-intercepts of the polynomial's graph, marking the locations where the curve crosses or touches the x-axis. Plotting these points is the first step in sketching the graph of the polynomial, as they provide a framework for understanding its behavior. The zeros divide the x-axis into intervals, within which the polynomial's value is either positive or negative. Understanding these intervals is crucial for accurately sketching the graph and for analyzing the polynomial's properties.

Sketching the Graph

Since the polynomial is of degree 4 (the highest power of x is 4), we know it has at most 3 turning points. The graph will cross the x-axis at each of the zeros, as they are distinct real roots. The end behavior of the polynomial is determined by the leading term, which in this case is $x^4$. As x approaches positive or negative infinity, $p(x)$ approaches positive infinity. This indicates that the graph rises on both ends.

The degree of the polynomial, 4, tells us that the graph can have up to 3 turning points, which are the points where the graph changes direction. The fact that the zeros are distinct real roots means that the graph will cross the x-axis at each zero, rather than just touching it and turning around. The end behavior of the polynomial, determined by the leading term $x^4$, is crucial for understanding how the graph behaves as x moves away from zero. In this case, the graph rises on both ends, indicating that it extends upwards in both the positive and negative directions. Combining this information with the location of the zeros allows for a reasonable sketch of the polynomial's graph.

Visualizing the Polynomial

The graph of the polynomial $p(x)$ will cross the x-axis at $x = -3$, $x = -2$, $x = -1$, and $x = 2$. It will have turning points between these zeros, and it will rise on both ends. The visual representation of the polynomial provides a clear understanding of its behavior, including its roots, turning points, and end behavior. The graph serves as a powerful tool for analyzing the polynomial's properties and for solving related problems. By visualizing the polynomial, we can gain insights into its characteristics and its relationship to the x-axis, making it easier to understand its mathematical properties and applications.

Conclusion

Finding the zeros of the polynomial $p(x) = (x^2 + 4x + 3)(x^2 - 4)$ involves factoring the polynomial, setting each factor to zero, and solving for x. The zeros are $x = -3$, $x = -1$, $x = 2$, and $x = -2$. These zeros represent the x-intercepts of the polynomial's graph and provide valuable information about its behavior. By understanding how to find the zeros of a polynomial, we can analyze its properties, sketch its graph, and solve related mathematical problems. This process is fundamental to algebra and calculus, with applications across various fields of science and engineering. Mastering the techniques for finding zeros is essential for anyone studying mathematics or its applications.