Polynomial

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Polynomials appear in a wide variety of areas within mathematics and science, including algebra, calculus, and numerical analysis. They are fundamental objects of study in algebra and are used to model various phenomena in physics, engineering, and economics.

Definition

A polynomial in a single variable \( x \) is an expression of the form:

\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \]

where \( a_n, a_{n-1}, \ldots, a_1, a_0 \) are constants called coefficients, and \( n \) is a non-negative integer called the degree of the polynomial. The term \( a_n x^n \) is called the leading term, and \( a_n \) is the leading coefficient. If \( a_n \neq 0 \), then \( n \) is the degree of the polynomial.

Types of Polynomials

Monomials, Binomials, and Trinomials

- **Monomial**: A polynomial with only one term, such as \( 5x^3 \). - **Binomial**: A polynomial with exactly two terms, such as \( x^2 - 4 \). - **Trinomial**: A polynomial with exactly three terms, such as \( x^3 + 2x - 7 \).

Homogeneous Polynomials

A polynomial is called homogeneous if all its terms have the same total degree. For example, \( 3x^2y + 2xy^2 \) is a homogeneous polynomial of degree 3.

Polynomial Functions

A polynomial function is a function that can be defined by evaluating a polynomial. For example, \( f(x) = 2x^3 - 3x + 1 \) is a polynomial function.

Operations on Polynomials

Addition and Subtraction

Polynomials can be added or subtracted by combining like terms. For example:

\[ (3x^2 + 2x + 1) + (x^2 - x + 4) = 4x^2 + x + 5 \]

Multiplication

The product of two polynomials is obtained by multiplying each term of one polynomial by each term of the other polynomial and then combining like terms. For example:

\[ (x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6 \]

Division

Polynomial division can be performed using long division or synthetic division. For example, dividing \( x^3 + 2x^2 - 5x + 6 \) by \( x - 2 \) using long division yields a quotient of \( x^2 + 4x + 3 \) and a remainder of 0.

Roots of Polynomials

The roots (or zeros) of a polynomial are the values of \( x \) for which the polynomial equals zero. For example, the roots of the polynomial \( x^2 - 5x + 6 \) are \( x = 2 \) and \( x = 3 \).

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-zero single-variable polynomial with complex coefficients has at least one complex root. This implies that a polynomial of degree \( n \) has exactly \( n \) roots, counted with multiplicity.

Polynomial Equations

A polynomial equation is an equation of the form \( P(x) = 0 \), where \( P(x) \) is a polynomial. Solving polynomial equations involves finding the roots of the polynomial.

Quadratic Equations

Quadratic equations are polynomial equations of degree 2 and can be solved using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \( ax^2 + bx + c = 0 \).

Higher-Degree Equations

For polynomials of degree higher than 2, various methods such as factoring, synthetic division, and numerical methods like Newton's method are used to find the roots.

Applications of Polynomials

Polynomials are used in numerous fields:

- **Physics**: Modeling motion and forces. - **Engineering**: Designing curves and surfaces. - **Economics**: Representing cost and revenue functions. - **Computer Science**: Algorithms for polynomial interpolation and approximation.