Real Zeros of Polynomials

Polynomials are algebraic expressions of terms with variables raised to non-negative integer powers. A central aspect of analyzing polynomial functions is determining their real zeros—values of the variable for which the polynomial evaluates to zero. These values represent the x-intercepts of the polynomial’s graph.

The Rational Zeros Theorem lists possible rational solutions for a polynomial equation with integer coefficients. If f(x)=a_nxⁿ+....+a_0​, then every rational zero is of the form p/q​, where p divides the constant term a₀,​ and q divides the leading coefficient a_n​. This theorem does not guarantee that each candidate is a zero, but it provides a finite set of rational numbers to test.

When synthetic division is used to check a value and the remainder is zero, that value is a zero of the polynomial. The polynomial can then be factored accordingly, reducing its degree and facilitating further analysis. Repeating this process simplifies the original polynomial to lower-degree factors.

Real zeros are essential in many scientific and engineering fields. In physics, they help analyze motion by identifying when an object changes direction or stops. In computer graphics, they are used for curve modeling and smooth animation transitions. Knowing how to find and interpret real zeros supports effective modeling, analysis, and problem-solving across diverse applications.