ZingPath: Polynomial Expressions and Factoring

A polynomial P(x) of degree n with real coefficients is a mathematical expression of the form are real numbers, , and n is a nonnegative integer. The term is called the leading term, whereas is called the constant term. Polynomials are classified in two ways: by the number of terms and by the degree of the polynomial.

Classification by number of terms:

·       monomial - a polynomial with one term (e.g., :4x , -1, 3x^4 )

·       binomial - a polynomial with two terms (e.g., : 4x-3, 3x^4 +x)

·       trinomial - a polynomial with three terms (e.g., : -3x^3-x+6, 3x^3 +x^2+6 )

·       polynomial - the general name for a polynomial with 1 or more terms (e.g., 3x^4+x+2x^3+x^2-4x-9, 3x^3+x^2+6)

Classification of polynomials by degree:

·       linear - a polynomial of degree one (e.g., : 2x)

·       quadratic - a polynomial of degree two (e.g., : x^2+4 )

·       cubic - a polynomial of degree three (e.g., : 3x^3+x^2+6)

·       quartic - a polynomial of degree four (e.g., : x4 - 5)

·       quintic - a polynomial of degree five (e.g., : x5 - 2x4 - 3x + 5)

Polynomials with a degree higher than three are also called by their numeric degree; for example, x^5 ? 2x^4 + 5x^3 +x^2 -x +6 can be referred to as a fifth degree polynomial.

Characteristics of polynomials.

Each polynomial has a leading term and a leading coefficient. Polynomials also have a constant term.

·       leading term - the term with the greatest exponent of x; for example, in the polynomial f(x) = 5x4 - x3 + 6x - 10, the leading term is 5x4.

·       leading coefficient - the number multiplying the variable in the leading term; in the polynomial f(x) = 5x4 - x3 + 6x - 10, the leading term is 5x4; the leading coefficient is 5.

·       constant term - a term with degree 0; in the example f(x) = 5x4 - x3 + 6x - 10, the term with degree zero is -10 since 10 = 10x0.

The constant term of a polynomial gives the y-intercept of the graph. Recall that the x-coordinate of any point along the y-axis is 0, so the y-intercept of a function occurs when x = 0. So, for a polynomial function, when evaluated at x = 0, we get

For example, the constant term of f(x) = x2 - 1 - 1, and the graph crosses the y-axis at (0, -1).

Standard form of a polynomial.

The standard form of a polynomial is . Any polynomial can be written in standard form by combining like terms and writing the terms in order of descending degree. For example, the standard form of 1-x+x^2-4x is given by -x^2-5x+1. The leading term, constant term, and degree of a polynomial are easy to read from the standard form.