The roots of an nth-degree polynomial are the n values (possibly complex and with multiplicity) that, when substituted for the variable, make the polynomial equal to zero.
Understanding Roots and Polynomials
A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. The degree of a polynomial is the highest power of the variable within the polynomial. For example:
f(x) = 3x^2 + 2x - 1is a polynomial of degree 2.g(x) = x^5 - 7x + 4is a polynomial of degree 5.h(x) = 4x - 9is a polynomial of degree 1.
The Fundamental Theorem of Algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. As a consequence, an nth-degree polynomial has exactly n complex roots, counting multiplicities. This means that some roots may be repeated.
Real vs. Complex Roots
- Real Roots: These are the values of x where the polynomial intersects the x-axis when graphed. They are real numbers.
- Complex Roots: These are roots that involve imaginary numbers (i.e., numbers that include the square root of -1, denoted as i). Complex roots always come in conjugate pairs (a + bi and a - bi) if the polynomial has real coefficients.
Finding the Roots
Finding the roots of a polynomial can range from simple to extremely complex, depending on its degree.
- Degree 1 (Linear): Solve by isolating the variable. Example:
2x + 3 = 0->x = -3/2 - Degree 2 (Quadratic): Use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2afor a polynomialax^2 + bx + c = 0. Factoring can also be used if applicable. - Degree 3 (Cubic) and Degree 4 (Quartic): There are formulas (Cardano's method and Ferrari's method, respectively), but they are often complex and cumbersome to use. Numerical methods are frequently employed.
- Degree 5 and Higher: The Abel-Ruffini theorem states that there is no general algebraic formula for finding the roots of polynomials of degree five or higher. Therefore, numerical methods (e.g., Newton-Raphson method, bisection method) are usually used to approximate the roots.
Multiplicity of Roots
A root can have a multiplicity greater than one. This means the same root appears multiple times. For example, the polynomial (x - 2)^2 = x^2 - 4x + 4 has a root of x = 2 with a multiplicity of 2. Graphically, a root with even multiplicity will "touch" the x-axis at that point but not cross it, while a root with odd multiplicity will cross the x-axis.
Example
Consider the polynomial f(x) = (x - 1)(x + 2)(x - 3) = x^3 - 2x^2 - 5x + 6. This is a 3rd-degree polynomial. Therefore, it has three roots: 1, -2, and 3.
Summary
An nth-degree polynomial has n roots, which can be real or complex, and may have multiplicities greater than one. The fundamental theorem of algebra guarantees the existence of these roots. While algebraic formulas exist for polynomials of degree 1 through 4, numerical methods are often necessary for higher-degree polynomials.
