What is the maximum number of zeros in a quadratic polynomial?

The maximum number of zeros a quadratic polynomial can have is two. This fundamental characteristic is directly related to the polynomial's degree.

Understanding Zeros in Quadratic Polynomials

A zero (also known as a root) of a polynomial is any value of the variable that makes the polynomial equal to zero. For a quadratic polynomial, which is an expression of the form ax² + bx + c (where a ≠ 0), the highest power of the variable x is 2. This means a quadratic polynomial has a degree of 2.

According to the Fundamental Theorem of Algebra, a polynomial will have a number of roots (zeros) equal to its degree in the complex number system. Therefore, a quadratic polynomial always has exactly two zeros when considering complex numbers.

Real vs. Complex Zeros

While a quadratic polynomial always has two zeros in the complex plane, these zeros can manifest as different types of real zeros:

• Two distinct real zeros: The graph of the quadratic polynomial (a parabola) intersects the x-axis at two separate points. This occurs when the discriminant (Δ = b² - 4ac) is positive (Δ > 0).
• One real zero (a repeated root): The parabola touches the x-axis at exactly one point, which is its vertex. This happens when the discriminant is zero (Δ = 0). In this case, both of the two complex roots are the same real number.
• No real zeros (two complex conjugate zeros): The parabola does not intersect the x-axis at all. This occurs when the discriminant is negative (Δ < 0), resulting in two complex (non-real) conjugate zeros.

Illustrative Examples of Quadratic Zeros

Let's examine examples that demonstrate the different numbers of real zeros a quadratic polynomial can have:

• Two Distinct Real Zeros:

  • Consider the polynomial: P(x) = x² - 5x + 6
  • Setting P(x) = 0 gives (x - 2)(x - 3) = 0.
  • The real zeros are x = 2 and x = 3. These are two distinct real zeros.

• One Real Zero (Repeated Root):

  • Consider the polynomial: P(x) = x² - 4x + 4
  • Setting P(x) = 0 gives (x - 2)² = 0.
  • The real zero is x = 2. This is a single, repeated real zero.

• No Real Zeros:

  • Consider the polynomial: P(x) = x² + 1
  • Setting P(x) = 0 gives x² = -1.
  • There are no real numbers whose square is -1. The zeros are x = i and x = -i (where i is the imaginary unit).

Summary of Quadratic Zero Scenarios

The number of real zeros for a quadratic polynomial can be summarized based on its discriminant:

In conclusion, while a quadratic polynomial always has two zeros in the complex number system, the maximum number of real zeros it can have is two, corresponding to the degree of the polynomial.