Finding Zeros Of Polynomials With Factors (x-3) And (x-7)

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Polynomial equations are a fundamental concept in algebra, and finding the values of variables that make a polynomial equal to zero, also known as finding the roots or zeros of the polynomial, is a crucial skill. This article will delve into the process of determining these values when the factors of the polynomial are given. Specifically, we'll address the question: Which values of x would make a polynomial equal to zero if the factors of the polynomial were (x - 3) and (x - 7)? We will explore the underlying principles, the step-by-step solution, and provide a comprehensive understanding of this concept.

Understanding the Zero Product Property

The cornerstone of solving this type of problem is the Zero Product Property. This property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. In mathematical terms, if a * b* = 0, then either a = 0 or b = 0 (or both). This seemingly simple property is the key to unlocking solutions for polynomial equations when they are expressed in factored form.

When dealing with polynomials, the factors often involve expressions containing the variable x. For instance, in our case, we have the factors (x - 3) and (x - 7). To find the values of x that make the polynomial equal to zero, we set each factor equal to zero and solve for x. This is because if either (x - 3) or (x - 7) equals zero, the entire product, and hence the polynomial, will equal zero.

Let's illustrate this with a simple example. Consider the equation (x - 2)(x + 1) = 0. According to the Zero Product Property, either (x - 2) = 0 or (x + 1) = 0. Solving these two equations gives us x = 2 and x = -1. These are the zeros or roots of the polynomial represented by the factored expression (x - 2)(x + 1). They are the values of x that make the polynomial equal to zero.

The Zero Product Property is not just a mathematical trick; it's a logical consequence of how multiplication works. Zero, when multiplied by any number, always results in zero. Therefore, to make a product equal to zero, at least one of the multiplicands must be zero. This principle applies regardless of the complexity of the factors, whether they are simple linear expressions like (x - 3) or more complex quadratic or higher-degree expressions.

In the context of polynomials, the roots or zeros are significant because they represent the x-intercepts of the polynomial's graph. These are the points where the graph crosses the x-axis. Understanding the zeros of a polynomial provides valuable information about its behavior and its relationship to the coordinate plane. Furthermore, finding the zeros is crucial in various applications of polynomials, such as solving equations, modeling real-world phenomena, and analyzing data.

In summary, the Zero Product Property is a powerful tool for solving polynomial equations in factored form. It allows us to break down a complex problem into simpler ones by setting each factor equal to zero. This property is not only essential for finding the zeros of polynomials but also forms the foundation for many other algebraic techniques.

Solving the Problem: Factors (x - 3) and (x - 7)

Now, let's apply the Zero Product Property to the specific problem at hand: Which values of x would make a polynomial equal to zero if the factors of the polynomial were (x - 3) and (x - 7)?

We are given that the factors of the polynomial are (x - 3) and (x - 7). This means the polynomial can be represented in factored form as P(x) = (x - 3)(x - 7), where P(x) denotes the polynomial. Our goal is to find the values of x that make P(x) equal to zero. In other words, we want to solve the equation (x - 3)(x - 7) = 0.

According to the Zero Product Property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x:

  1. Set the first factor equal to zero:
    • x - 3 = 0
  2. Solve for x:
    • Add 3 to both sides of the equation:
      • x = 3

So, one solution is x = 3. This means that if we substitute x = 3 into the polynomial, the factor (x - 3) will become zero, and the entire polynomial will equal zero.

Next, we do the same for the second factor:

  1. Set the second factor equal to zero:
    • x - 7 = 0
  2. Solve for x:
    • Add 7 to both sides of the equation:
      • x = 7

Thus, the second solution is x = 7. Substituting x = 7 into the polynomial will make the factor (x - 7) zero, resulting in the entire polynomial being zero.

Therefore, the values of x that make the polynomial equal to zero are x = 3 and x = 7. These are the zeros or roots of the polynomial defined by the factors (x - 3) and (x - 7). Geometrically, these values represent the points where the graph of the polynomial intersects the x-axis. If we were to graph the polynomial P(x) = (x - 3)(x - 7), we would find that the graph crosses the x-axis at x = 3 and x = 7.

In summary, by applying the Zero Product Property, we were able to systematically find the zeros of the polynomial by setting each factor equal to zero and solving for x. This method is a fundamental technique in algebra and is widely used for solving polynomial equations in factored form.

Analyzing the Options

Now that we have solved the problem and determined that the values of x that make the polynomial equal to zero are 3 and 7, let's analyze the given options to identify the correct answer.

The original question presented four options:

A. 3 and 7 B. 3 and -7 C. -3 and 7 D. -3 and -7

We found that the solutions to the equation (x - 3)(x - 7) = 0 are x = 3 and x = 7. Let's examine each option in light of this result:

  • Option A: 3 and 7 - This option matches our solution exactly. When x is 3, the factor (x - 3) becomes (3 - 3) = 0. When x is 7, the factor (x - 7) becomes (7 - 7) = 0. Thus, this option is correct.

  • Option B: 3 and -7 - This option includes 3, which is a correct solution. However, it also includes -7. If we substitute x = -7 into the factor (x - 7), we get (-7 - 7) = -14, which is not zero. Therefore, this option is incorrect.

  • Option C: -3 and 7 - This option includes 7, which is a correct solution. However, it also includes -3. If we substitute x = -3 into the factor (x - 3), we get (-3 - 3) = -6, which is not zero. Therefore, this option is incorrect.

  • Option D: -3 and -7 - This option includes neither 3 nor 7. Substituting x = -3 into (x - 3) gives -6, and substituting x = -7 into (x - 7) gives -14. Neither value makes the factors zero, so this option is incorrect.

Based on our analysis, only Option A: 3 and 7 accurately represents the values of x that make the polynomial equal to zero when the factors are (x - 3) and (x - 7). The other options include incorrect values that do not satisfy the Zero Product Property.

This process of analyzing the options after solving the problem is a valuable strategy for verifying your answer and ensuring accuracy. It helps to confirm that the solutions you found are indeed the correct ones and that no mistakes were made in the solving process. In multiple-choice questions, carefully evaluating each option against your solution is a crucial step in maximizing your chances of selecting the correct answer.

Generalizing the Concept

The principle we've used to solve this specific problem can be generalized to any polynomial expressed in factored form. The key takeaway is that to find the zeros of a polynomial in factored form, you set each factor equal to zero and solve for the variable. This method stems directly from the Zero Product Property, which is a fundamental concept in algebra.

Consider a general polynomial P(x) that can be factored as P(x) = (x - a)(x - b)(x - c)..., where a, b, c, and so on are constants. To find the zeros of this polynomial, we set P(x) = 0:

(x - a)(x - b)(x - c)... = 0

Applying the Zero Product Property, we know that at least one of the factors must be equal to zero. Therefore, we set each factor equal to zero and solve for x:

  • x - a = 0 => x = a
  • x - b = 0 => x = b
  • x - c = 0 => x = c
  • ...

This shows that the zeros of the polynomial are x = a, x = b, x = c, and so on. In other words, the zeros of the polynomial are the values that make each individual factor equal to zero.

This generalized approach is applicable to polynomials with any number of factors. For example, if a polynomial has three factors, we set each of the three factors equal to zero and solve for x. If a polynomial has ten factors, we repeat the process for all ten factors. The Zero Product Property ensures that we capture all possible solutions by considering each factor individually.

Moreover, this concept extends beyond linear factors of the form (x - a). If a polynomial has factors like (2x + 1) or (x^2 - 4), we still apply the same principle. We set each factor equal to zero and solve for x, which may involve solving linear, quadratic, or higher-degree equations depending on the complexity of the factors.

Understanding this generalized approach is crucial for solving a wide range of polynomial equations. It provides a systematic method for finding the zeros of polynomials in factored form, regardless of the number or complexity of the factors. This skill is fundamental in algebra and is essential for tackling more advanced topics in mathematics and related fields.

Conclusion

In conclusion, we have successfully determined the values of x that make a polynomial equal to zero when its factors are given as (x - 3) and (x - 7). By applying the Zero Product Property, we found that the solutions are x = 3 and x = 7, corresponding to option A. This problem highlights the importance of understanding and applying the Zero Product Property, which is a cornerstone of solving polynomial equations in factored form.

We have also extended this concept to a more general case, demonstrating that the zeros of a polynomial in factored form can be found by setting each factor equal to zero and solving for the variable. This approach is universally applicable to polynomials with any number of factors, regardless of their complexity. Mastering this technique is essential for success in algebra and beyond.

The ability to find the zeros of a polynomial is not just a mathematical exercise; it has significant applications in various fields. For example, in physics, the zeros of a polynomial can represent the equilibrium points of a system. In engineering, they can correspond to the resonant frequencies of a circuit. In economics, they can indicate the break-even points for a business. Understanding how to find these zeros allows us to model and analyze real-world phenomena effectively.

Therefore, the skills and concepts discussed in this article are not only valuable for solving specific problems but also for developing a deeper understanding of mathematical principles and their applications in diverse contexts. By grasping the fundamental concepts and practicing problem-solving techniques, you can build a strong foundation in algebra and unlock new possibilities in your academic and professional pursuits. Remember, the key to success in mathematics lies in understanding the underlying principles and applying them consistently and accurately.