Section 7.1 An Introduction to Polynomials

Terminology A monomial is numeral, a variable, or the product of a numeral and one or more values. Monomials with no variables are called constants. A coefficient is the numerical factor in a monomial. The degree of a monomial is the sum of the exponents of its variables.

Terminology A polynomial is a monomial or a sum of terms that are monomials. Polynomials can be classified by the number of terms they contain. A polynomial with two terms is binomial. A polynomial with three terms is a trinomial. The degree of a polynomial is the same as that of its term with the greatest degree.

Classification of a Polynomial By Degree Degree Name Example n = 0constant 3 n = 1linear 5x + 4 n = 2quadratic-x² + 11x – 5 n = 3cubic 4x³ - x² + 2x – 3 n = 4quartic 9x⁴ + 3x³ + 4x² - x + 1 n = 5 quintic -2x⁵ + 3x⁴ - x³ + 3x² - 2x + 6

Classification of Polynomials 2x ³ - 3x + 4x⁵-2x³ + 3x⁴ + 2x³ + 5 The degree is 5The degree is 4 Quintic TrinomialQuartic Binomial x² + 4 – 8x – 2x³3x³ + 2 – x³ - 6x⁵ The degree is 3The degree is 5 Cubic PolynomialQuintic Trinomial

Adding and Subtracting Polynomials The standard form of a polynomial expression is written with the exponents in descending order of degree. (-2x² - 3x³ + 5x + 4) + (-2x³ + 7x – 6) - 5x³ - 2x² + 12x – 2 (3x³ - 12x² - 5x + 1) – (-x² + 5x + 8) (3x³ - 12x² - 5x + 1) + (x² - 5x – 8) 3x³ - 11x² - 10x - 7

Graphing Polynomial Functions A polynomial function is a function that is defined by a polynomial expression. Graph f(x) = 3x³ - 5x² - 2x +1 Describe its general shape.

Section 7.2 Polynomial Functions and Their Graphs

Graphs of Polynomial Functions When a function rises and then falls over an interval from left to right, the function has a local maximum. f(a) is a local maximum (plural, local maxima) if there is an interval around a such that f(a) > f(x) for all values of x in the interval, where x ≠ a. If the function falls and then rises over an interval from left to right, it has a local minimum. f(a) is a local minimum (plural, local minima) if there is an interval around a such that f(a) < f(x) for all values of x in the interval, where x ≠ a.

Graphs of Polynomial Functions The points on the graph of a polynomial function that correspond to local maxima and local minima are called turning points. Functions change from increasing to decreasing or from decreasing to increasing at turning points. A cubic function has at most 2 turning points, and a quartic function has at most 3 turning points. In general, a polynomial function of degree n has at most n – 1 turning points.

Increasing and Decreasing Functions Let x₁ and x₂ be numbers in the domain of a function, f. The function f is increasing over an open interval if for every x₁ < x₂ in the interval, f(x₁) < f(x₂). The function f is decreasing over an open interval if for every x₁ f(x₂).

Continuity of a Polynomial Function Every polynomial function y = P(x) is continuous for all values of x. Polynomial functions are one type of continuous functions. The graph of a continuous function is unbroken. The graph of a discontinuous function has breaks or holes in it.

If a polynomial function is written in standard form f(x) = a x ⁿ + a xⁿ⁻¹ + · · · + a₁x + a₀, ⁿ ⁿ⁻¹ The leading coefficient is a. ⁿ The leading coefficient is the coefficient of the term of greatest degree in the polynomial.

Section 7.3 Products and Factors of Polynomials