Polynomial Definition

Definition (Polynomial Definition) Polynomial Expression. A polynomial in is an algebraic expression that can be written in the form



where is a nonnegative integer, is a variable, and each of the numbers are called the coefficients of the polynomials.If we say this is a polynomial expression of degree is known as the leading coefficient, and is called the constant coefficient.

Example (Polynomial Definition) Polynomial Expression. Here are some common examples of polynomials expressions:   and There following expressions are not polynomial expressions: and

Definition (Polynomial Definition) Polynomial Function. Any function that has the form



where is a nonnegative integer and   are real numbers is called polynomial function. If we say is a polynomial function of degree is known as the leading coefficient, and is called the constant coefficient.

Example (Polynomial Definition) Polynomial Function. Linear functions have the form quadratic functions have the form cubic polynomial functions have the form . A quartic function



has leading coefficient constant coefficient and degree 4.

Definition (Polynomial Definition) Polynomial Equation. A polynomial equation in one variable is an equation that can be written in the form



where is a nonnegative integer and   are real numbers. If we say we have an equation of degree is known as the leading coefficient, and is called the constant coefficient.

Example (Polynomial Definition) Polynomial Equation. Verify that is a solution to the polynomial equation
    
    Solution. By substituting into the polynomial equation we find,
    


which verifies that is a solution to the equation   

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