Finding Zeros Of Polynomial P(x)=(x^2-9)(x^2+x-2) A Step-by-Step Guide

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In the realm of mathematics, polynomials play a crucial role. They are fundamental building blocks in algebra and calculus, and understanding their behavior is essential for solving various problems. One of the key aspects of analyzing polynomials is finding their zeros, also known as roots or solutions. These are the values of x for which the polynomial evaluates to zero. In this comprehensive guide, we will delve into the process of finding the zeros of a polynomial, using the example of the polynomial p(x) = (x² - 9)(x² + x - 2). Our aim is to provide a clear, step-by-step approach that will empower you to tackle similar problems with confidence. Let's embark on this mathematical journey together and unravel the mysteries of polynomial zeros.

Understanding Polynomial Zeros

Polynomial zeros are the values of the variable x that make the polynomial equal to zero. These zeros are also known as roots or solutions of the polynomial equation p(x) = 0. Finding the zeros of a polynomial is a fundamental task in algebra and has wide-ranging applications in various fields, including engineering, physics, and economics. In essence, polynomial zeros represent the points where the graph of the polynomial intersects the x-axis. These points hold significant information about the polynomial's behavior, such as its intervals of increase and decrease, its turning points, and its overall shape. By understanding the zeros of a polynomial, we gain valuable insights into its properties and can use this knowledge to solve real-world problems.

Why are Polynomial Zeros Important?

The zeros of a polynomial provide crucial information about its behavior and properties. They are the points where the graph of the polynomial intersects the x-axis, and they reveal a great deal about the polynomial's structure. For instance, the number of zeros (counting multiplicity) corresponds to the degree of the polynomial, a fundamental concept in algebra. Moreover, the zeros can be used to factor the polynomial, which simplifies many algebraic manipulations and makes it easier to solve equations involving the polynomial. In practical applications, polynomial zeros arise in various contexts. In engineering, they can represent the resonant frequencies of a system or the equilibrium points of a dynamic system. In economics, they can indicate the break-even points of a business or the points of market equilibrium. In physics, they can describe the stable states of a physical system or the solutions to certain differential equations. Therefore, understanding how to find polynomial zeros is essential for anyone working with mathematical models in any field.

Factoring the Polynomial

To find the zeros of the polynomial p(x) = (x² - 9)(x² + x - 2), the first step is to factor it completely. Factoring a polynomial involves expressing it as a product of simpler polynomials, which makes it easier to identify the zeros. This is because if a product of factors is zero, then at least one of the factors must be zero. In our case, we have two factors: (x² - 9) and (x² + x - 2). Let's factor each of these individually.

Factoring (x² - 9)

The first factor, (x² - 9), is a difference of squares, which can be factored using the formula a² - b² = (a - b)(a + b). Here, a = x and b = 3, so we have:

x² - 9 = (x - 3)(x + 3)

This factorization is crucial because it immediately reveals two of the polynomial's zeros: x = 3 and x = -3. These are the values of x that make the factor (x - 3) or (x + 3) equal to zero, and consequently, make the entire polynomial p(x) equal to zero. Understanding and recognizing patterns like the difference of squares is a valuable skill in factoring polynomials and finding their zeros.

Factoring (x² + x - 2)

The second factor, (x² + x - 2), is a quadratic trinomial. To factor this, we need to find two numbers that multiply to -2 and add to 1 (the coefficient of the x term). These numbers are 2 and -1. Therefore, we can factor the trinomial as follows:

x² + x - 2 = (x + 2)(x - 1)

This factorization gives us two more zeros of the polynomial: x = -2 and x = 1. These are the values of x that make the factors (x + 2) or (x - 1) equal to zero, and consequently, make the polynomial p(x) equal to zero. Factoring quadratic trinomials is a common technique in algebra, and mastering it is essential for finding the zeros of polynomials of higher degrees.

Complete Factorization

Now that we have factored both factors, we can write the complete factorization of the polynomial p(x):

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

This complete factorization reveals all the zeros of the polynomial, which are the values of x that make any of the factors equal to zero. In the next section, we will explicitly state these zeros.

Finding the Zeros

Now that we have the completely factored form of the polynomial, p(x) = (x - 3)(x + 3)(x + 2)(x - 1), finding the zeros becomes straightforward. The zeros are the values of x that make p(x) = 0. This occurs when any of the factors are equal to zero. We simply set each factor equal to zero and solve for x.

Setting Factors to Zero

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

Therefore, the zeros of the polynomial p(x) = (x² - 9)(x² + x - 2) are x = 3, x = -3, x = -2, and x = 1. These are the points where the graph of the polynomial intersects the x-axis, and they represent the solutions to the equation p(x) = 0. Understanding how to find these zeros is a fundamental skill in algebra and calculus, and it has numerous applications in various fields.

Verifying the Zeros

To ensure our calculations are correct, it's always a good practice to verify the zeros we found. We can do this by substituting each zero back into the original polynomial p(x) and confirming that the result is indeed zero. Let's verify each zero:

  • p(3) = (3² - 9)(3² + 3 - 2) = (0)(10) = 0
  • p(-3) = ((-3)² - 9)((-3)² + (-3) - 2) = (0)(4) = 0
  • p(-2) = ((-2)² - 9)((-2)² + (-2) - 2) = (-5)(0) = 0
  • p(1) = (1² - 9)(1² + 1 - 2) = (-8)(0) = 0

As we can see, substituting each zero into the polynomial results in zero, confirming that our zeros are correct. This verification step is crucial in catching any potential errors and ensuring the accuracy of our results.

Conclusion

In this guide, we have successfully found the zeros of the polynomial p(x) = (x² - 9)(x² + x - 2). We started by understanding the importance of polynomial zeros and their significance in various applications. Then, we factored the polynomial completely, utilizing techniques such as the difference of squares and factoring quadratic trinomials. Finally, we set each factor equal to zero and solved for x to find the zeros: x = 3, x = -3, x = -2, and x = 1. We also verified our results by substituting each zero back into the original polynomial and confirming that it evaluates to zero.

This process demonstrates a systematic approach to finding the zeros of polynomials, which can be applied to a wide range of polynomial functions. By mastering these techniques, you will gain a deeper understanding of polynomial behavior and be well-equipped to tackle more complex problems in algebra and calculus. Remember, practice is key to success in mathematics, so continue to explore and solve various polynomial problems to solidify your understanding. The world of polynomials is vast and fascinating, and finding their zeros is just one step in unlocking their secrets.