Section 7.1

1. Section 7.1 An Introduction to Polynomials

2. 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.

3. 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.

4. Classification of a Polynomial By Degree Degree Name Example n = 0 constant 3 n = 1linear 5x + 4 n = 2quadratic-x² + 11x – 5 n = 3cubic 4x³ - x² + 2x – 3 n = 4quartic9x⁴ + 3x³ + 4x² - x + 1 n = 5 quintic-2x⁵ + 3x⁴ - x³ + 3x² - 2x + 6

5. Classification of Polynomials • 2x³ - 3x + 4x⁵ -2x³ + 3x⁴ + 2x³ + 5 • The degree is 5 The degree is 4 • Quintic Trinomial Quartic Binomial • x² + 4 – 8x – 2x³ 3x³ + 2 – x³ - 6x⁵ • The degree is 3 The degree is 5 • Cubic Polynomial Quintic Trinomial

6. 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

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.

8. Section 7.2 Polynomial Functions and Their Graphs

9. 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.

10. 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.

11. 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 decreasingover an open interval if for every x₁ < x₂ in the interval, f(x₁) > f(x₂).

12. 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.

13. If a polynomial function is written in standard form • f(x) = a xⁿ + 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.

14. Section 7.3 Products and Factors of Polynomials