Student Tutorial: Polynomial Concepts: Definitions

Instructions:

Study these polynomial definitions.

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Polynomial. An expression made up of one or more monomials. The degree of the polynomial is determined by the highest exponent.

Degree of a Polynomial. The value of the greatest exponent for a polynomial written in standard form.

Monomial. An expression made up of a product of numbers and one or more variables raised to non-negative integer exponents. A polynomial with one term.

Binomial. A polynomial expression consisting of two terms.

Trinomial. A polynomial expression with three terms.

Polynomial Addition. Polynomials are closed under addition, meaning that the sum of two or more polynomials is a polynomial.

Polynomial Subtraction. Polynomials are closed under subtraction, meaning that the difference of two polynomials is another polynomial.

Polynomial Multiplication. Polynomials are closed under multiplication, meaning that the product of two ore more polynomials is another polynomial.

Synthetic Division. A method for finding the linear factors (if any) of a polynomial.

Polynomial Expansion. To use the distributive property to write a polynomial in factored form into standard form.

Factored Polynomial. Expressing a polynomial as the product of two or more lower-degree polynomial terms.

Factored Quadratic. Expressing a polynomial of degree 2 as the product of one or two linear terms.

Factored Cubic. Expressing a polynomial of degree 3 as the product of linear and/or quadratic terms.

Difference of Squares. When a quadratic polynomial is written as a difference of squares, it can be factored as shown.

Difference of Cubes. When a cubic polynomial is written as a difference of cubes, it can be factored as shown.

Sum of Cubes. When a cubic polynomial is written as a sum of cubes, it can be factored as shown.

Binomial Factor. A polynomial P(x), of degree n, is said to have a binomial factor (x – a), if it can be written as the product of this factor and another polynomial of degree n – 1.

Factor Theorem. Given polynomial P(x), if P(a) = 0 for some number a, then (x – a) is a linear factor of P(x). Likewise, if (x – a) is a linear factor of P(x), then P(a) = 0.