Which Function Has Zeros At X 10 And X 2

Which Function Has Zeros at x = 10 and x = 2? Exploring Polynomial Functions and Beyond

Finding functions with specific zeros is a fundamental concept in algebra and calculus. This article delves deep into the question: Which function has zeros at x = 10 and x = 2? We'll explore various types of functions, from simple polynomials to more complex scenarios, demonstrating how to construct functions that meet this criterion and highlighting the underlying mathematical principles.

Understanding Zeros of a Function

Before diving into specific examples, let's clarify what we mean by "zeros" of a function. A zero (also known as a root or x-intercept) of a function f(x) is a value of x for which f(x) = 0. Graphically, zeros represent the points where the graph of the function intersects the x-axis.

Finding the zeros of a function is crucial for understanding its behavior, solving equations, and modeling real-world phenomena.

Polynomial Functions: A Simple Approach

The simplest type of function with defined zeros is a polynomial function. For a function to have zeros at x = 10 and x = 2, we can construct a polynomial using the factor theorem. This theorem states that if 'a' is a zero of a polynomial, then (x - a) is a factor of the polynomial.

Therefore, if our function has zeros at x = 10 and x = 2, it must have factors (x - 10) and (x - 2). The simplest such polynomial is:

f(x) = (x - 10)(x - 2)

Expanding this expression, we get:

f(x) = x² - 12x + 20

This quadratic function has zeros at x = 10 and x = 2. We can verify this by setting f(x) = 0 and solving for x:

x² - 12x + 20 = 0

(x - 10)(x - 2) = 0

x = 10 or x = 2

Exploring Higher-Order Polynomials

While the quadratic function above is the simplest solution, infinitely many other polynomial functions can also have zeros at x = 10 and x = 2. We can simply multiply our original function by any other polynomial:

g(x) = (x - 10)(x - 2) * h(x)

where h(x) can be any polynomial. For example:

• g(x) = (x - 10)(x - 2)(x + 5) This cubic polynomial has zeros at x = 10, x = 2, and x = -5.
• g(x) = (x - 10)(x - 2)(x² + 1) This quartic polynomial has zeros at x = 10 and x = 2, and two complex zeros (because x² + 1 = 0 has no real solutions).

The introduction of additional factors introduces more zeros, but the original zeros at x = 10 and x = 2 remain.

Beyond Polynomials: Other Function Types

While polynomial functions are a straightforward approach, other types of functions can also possess specified zeros. Let's explore a few examples:

Rational Functions

Rational functions are functions of the form f(x) = p(x) / q(x), where p(x) and q(x) are polynomials. A rational function will have a zero at x = a if and only if p(a) = 0 and q(a) ≠ 0.

We can create a rational function with zeros at x = 10 and x = 2 by constructing a numerator with (x - 10)(x - 2) as a factor and ensuring the denominator doesn't have zeros at those points. For example:

f(x) = (x - 10)(x - 2) / (x + 1)

Trigonometric Functions

Trigonometric functions such as sine and cosine are periodic and have infinitely many zeros. While we cannot directly construct a simple trigonometric function with precisely zeros at only x = 10 and x = 2, we can use transformations and combinations to create more complex functions that include these zeros. This often involves more advanced techniques.

Exponential and Logarithmic Functions

Exponential and logarithmic functions also generally do not have finite zeros. Their behaviour is significantly different from polynomials. While we could potentially construct complex functions involving these that incorporate the desired zeros, it would require intricate manipulations and likely be less intuitive than polynomial approaches.

The Importance of Multiplicity

When considering zeros, the concept of multiplicity is important. The multiplicity of a zero is the number of times the corresponding factor appears in the factored form of the function. For example:

f(x) = (x - 10)²(x - 2)

In this case, x = 10 has a multiplicity of 2, while x = 2 has a multiplicity of 1. The multiplicity affects the behavior of the function near the zero; a higher multiplicity implies a flatter curve near the x-axis at that point.

Applications and Real-World Examples

The ability to construct functions with specified zeros is crucial in many applications, including:

• Modeling physical phenomena: Many real-world processes can be modeled using functions with specific zeros representing critical points or equilibrium states.
• Signal processing: In signal processing, zeros of a function's transfer function correspond to frequencies that are suppressed or attenuated.
• Engineering design: Zeroes are important in determining stability, resonance frequencies, and other critical parameters in engineering systems.
• Computer graphics: Functions with specific zeros can be used to generate curves and shapes with precise properties.

Conclusion: A Multifaceted Problem

The question, "Which function has zeros at x = 10 and x = 2?" has no single answer. We've demonstrated that countless functions, primarily polynomial but also including rational functions, and potentially more complex functions, can satisfy this condition. The choice of function depends on the specific application and the desired properties of the function beyond simply having zeros at those two points. Understanding the underlying principles of zero finding and the behavior of different function types is key to successfully constructing and interpreting these functions in various mathematical and real-world contexts. The exploration of multiplicity and the potential inclusion of complex zeros further expands the range of possibilities. This article has provided a foundational understanding of the methods and concepts involved in solving this problem and similar ones, equipping readers to tackle similar problems involving the determination of functions with specific zero values.