Polynomial Roots — Definition, Formula & Examples
Polynomial roots are the values of the variable that make the polynomial equal to zero. They are also called zeros or solutions of the polynomial.
A root of a polynomial is a value such that . By the Fundamental Theorem of Algebra, a polynomial of degree with complex coefficients has exactly roots in the complex numbers, counted with multiplicity.
Key Formula
Where:
- = A polynomial function
- = A root (value that makes the polynomial equal zero)
How It Works
To find the roots of a polynomial, you set it equal to zero and solve for the variable. For quadratics, you can factor, complete the square, or use the quadratic formula. For higher-degree polynomials, factoring and the Rational Root Theorem are common strategies. If is a root of , then is a factor of , and vice versa — this connection between roots and factors is central to working with polynomials.
Worked Example
Problem: Find the roots of the polynomial .
Set equal to zero: Write the equation to solve.
Factor: Find two numbers that multiply to 6 and add to −5. Those numbers are −2 and −3.
Solve each factor: Set each factor equal to zero and solve.
Answer: The roots are and .
Why It Matters
Finding polynomial roots is essential in Algebra 2, Precalculus, and Calculus — you need roots to factor expressions, sketch graphs, and solve real-world equations. Engineers use polynomial roots to analyze stability in control systems, and physicists use them to solve equations of motion.
Common Mistakes
Mistake: Confusing a root with a factor. Students sometimes say "the root is " instead of "the root is ."
Correction: A root is a number (), while the corresponding factor is a binomial expression . If is a root, then is the factor.