Which Of These Functions Has Exactly Two Different Zeros

Which of These Functions Has Exactly Two Different Zeros? A Deep Dive into Polynomial Roots

Finding the zeros (or roots) of a function is a fundamental concept in algebra and calculus. Many functions, particularly polynomials, can have multiple zeros, some with multiplicities greater than one. This article will explore different types of functions and delve into how to determine which functions have exactly two distinct zeros. We'll focus on methods for identifying these functions and provide examples to solidify our understanding.

Understanding Zeros and Roots

Before we dive into specific functions, let's clarify the terminology. A zero of a function f(x) is a value of x for which f(x) = 0. This is also known as a root of the function. The zeros represent the points where the graph of the function intersects the x-axis.

A function can have multiple zeros, and these zeros can be real or complex (involving imaginary numbers). A zero can also have a multiplicity, which indicates how many times the zero is repeated. For instance, if a function has a factor of (x-2)² , then x = 2 is a zero with multiplicity 2.

Types of Functions and Their Zeros

Several types of functions are commonly encountered, each with its own characteristics regarding the number and nature of their zeros:

1. Linear Functions

A linear function has the form f(x) = mx + c, where m and c are constants. A linear function has exactly one zero, which is found by setting f(x) = 0 and solving for x: x = -c/m (provided m ≠ 0).

2. Quadratic Functions

A quadratic function is of the form f(x) = ax² + bx + c, where a, b, and c are constants, and a ≠ 0. A quadratic function can have either:

• Two distinct real zeros: This occurs when the discriminant (b² - 4ac) is positive.
• One real zero (with multiplicity 2): This occurs when the discriminant is zero. The zero is given by x = -b/(2a).
• Two complex conjugate zeros: This occurs when the discriminant is negative. These zeros are of the form α ± βi, where α and β are real numbers and i is the imaginary unit.

Example: Consider the quadratic function f(x) = x² - 5x + 6. Factoring this gives (x-2)(x-3) = 0, revealing zeros at x = 2 and x = 3. This function has exactly two distinct real zeros.

3. Cubic Functions

Cubic functions have the form f(x) = ax³ + bx² + cx + d, where a, b, c, and d are constants, and a ≠ 0. A cubic function will always have at least one real zero. It can have:

• Three distinct real zeros.
• One real zero and two complex conjugate zeros.
• One real zero (with multiplicity 3).
• One real zero (with multiplicity 1) and one real zero (with multiplicity 2).

Example: The function f(x) = x³ - 6x² + 11x - 6 can be factored as (x-1)(x-2)(x-3) = 0, demonstrating three distinct real zeros.

4. Polynomial Functions of Higher Degree

Polynomials of degree n (where n is a positive integer) can have at most n zeros (counting multiplicities). The number of distinct real zeros can vary widely. Finding the zeros of higher-degree polynomials often requires numerical methods or more advanced techniques. The Fundamental Theorem of Algebra guarantees that a polynomial of degree n has exactly n complex zeros (counting multiplicities).

5. Trigonometric Functions

Trigonometric functions, such as sin(x), cos(x), and tan(x), are periodic and have infinitely many zeros. Therefore, they do not have exactly two distinct zeros.

6. Exponential and Logarithmic Functions

Exponential functions (eˣ, aˣ) and logarithmic functions (ln(x), logₐ(x)) generally have only one zero or no zeros at all. For example, eˣ has no real zeros, and ln(x) has one zero at x = 1.

Identifying Functions with Exactly Two Distinct Zeros

To identify a function with exactly two distinct zeros, we need to consider the following:

• The degree of the polynomial: While higher-degree polynomials can have exactly two distinct zeros, this often requires specific coefficients and careful consideration of potential factors.
• The nature of the zeros: We must ensure the zeros are distinct (not repeated) and real (not complex).
• Factoring and the quadratic formula: Factoring the polynomial is the most straightforward approach to determine the zeros. If this is not easily done, the quadratic formula can be applied (if it's a quadratic function) to find the roots.

Example: Let's say we have the function f(x) = x⁴ - 5x² + 4. This is a quartic function, but it can be treated as a quadratic in x²: Let y = x². Then, the equation becomes y² - 5y + 4 = 0. This factors as (y-1)(y-4) = 0, giving y = 1 and y = 4. Substituting back, we get x² = 1 and x² = 4. This yields four zeros: x = ±1 and x = ±2. Therefore, this function has exactly four distinct zeros.

Strategies for Finding Functions with Exactly Two Zeros:

1. Start with a quadratic: The simplest way to ensure exactly two distinct zeros is to construct a quadratic function with a positive discriminant (b² - 4ac > 0). This guarantees two distinct real roots.
2. Higher-degree polynomials require careful construction: Creating higher-degree polynomials with precisely two distinct real zeros requires factoring techniques or sophisticated approaches to guarantee only two unique roots. This often involves ensuring that any repeated roots are complex or that higher-order factors don't introduce additional real roots.
3. Graphical Analysis: Graphing the function allows for visual confirmation of the number and nature of the zeros.

Conclusion

Determining which functions possess exactly two distinct zeros requires a thorough understanding of function types, factoring techniques, and the properties of roots. While quadratic functions provide the most straightforward path to achieving this outcome, constructing higher-degree polynomials with exactly two distinct zeros demands careful planning and manipulation of coefficients to avoid introducing additional real roots. Combining algebraic methods with graphical analysis offers a robust approach to identifying and verifying functions with the desired characteristic. Remember that the nature of the roots (real or complex) plays a crucial role in determining the total number of distinct zeros.