1. Copyright © Cengage Learning. All rights reserved. Polynomials 4

2. Copyright © Cengage Learning. All rights reserved. Section 4.4 Polynomials

3. 3 Objectives Determine whether an expression is a polynomial. Classify a polynomial as a monomial, binomial, or trinomial, if applicable. Find the degree of a polynomial. Evaluate a polynomial. 1 1 2 2 3 3 4 4

4. 4 Objectives Evaluate a polynomial function. Graph a linear, quadratic, and cubic polynomial function. 6 6 5 5

5. 5 Polynomials In algebra, exponential expressions may be combined to form polynomials. In this section, we will introduce the topic of polynomials and graph some basic polynomial functions.

6. 6 Determine whether an expression is a polynomial 1.

7. 7 Determine whether an expression is a polynomial Recall that expressions such as 3x 4y 2 –8x 2 y 3 and 25 with constant and/or variable factors are called algebraic terms. The coefficients of the first three of these terms are 3, 4, and –8, respectively. Because 25 = 25x 0, 25 is referred to as a constant.

8. 8 Determine whether an expression is a polynomial Polynomials A polynomial is an algebraic expression that is a single term or the sum of several terms containing whole-number exponents on the variables. Here are some examples of polynomials: 8xy 2 t 3x + 2 4y 2 – 2y + 3 and 3a – 4b – 4c – 8d Comment The expression 2x 3 – 3y –2 is not a polynomial, because the second term contains a negative exponent on a variable base.

9. 9 Example Determine whether each expression is a polynomial. a. x 2 + 2x + 1 b. 3x –1 – 2x – 3 c. x 3 – 2.3x + 5 d. –2x + 3x 1/2 A polynomial. No. The first term has a negative exponent on a variable base. A polynomial No. The second term has a fractional exponent on a variable base.

10. 10 Classify a polynomial as a monomial, binomial, or trinomial, if applicable 2.

11. 11 Classify a polynomial as a monomial, binomial, or trinomial, if applicable A polynomial with one term is called a monomial. A polynomial with two terms is called a binomial. A polynomial with three terms is called a trinomial. Here are some examples.

12. 12 Example Classify each polynomial as a monomial, a binomial, or a trinomial, if applicable. a. 5x 4 + 3x b. 7x 4 – 5x 3 – 2 c. –5x 2 y 3 d. 9x 5 – 5x 2 + 8x – 7 Since the polynomial has two terms, it is a binomial. Since the polynomial has three terms, it is a trinomial. Since the polynomial has one term, it is a monomial. Since the polynomial has four terms, it has no special name. It is none of these.

13. 13 Find the degree of a polynomial 3.

14. 14 Find the degree of a polynomial The monomial 7x 6 is called a monomial of sixth degree or a monomial of degree 6, because the variable x occurs as a factor six times. The monomial 3x 3 y 4 is a monomial of the seventh degree, because the variables x and y occur as factors a total of seven times. Other examples are –2x 3 is a monomial of degree 3. 47x 2 y 3 is a monomial of degree 5. 18x 4 y 2 z 8 is a monomial of degree 14. 8 is a monomial of degree 0, because 8 = 8x 0.

15. 15 Find the degree of a polynomial Degree of a Monomial If a is a nonzero coefficient, the degree of the monomial ax n is n. The degree of a monomial with several variables is the sum of the exponents on those variables. Comment Note that the degree of ax n is not defined when a = 0. Since ax n = 0 when a = 0, the constant 0 has no defined degree.

16. 16 Find the degree of a polynomial Because each term of a polynomial is a monomial, we define the degree of a polynomial by considering the degree of each of its terms. Degree of a Polynomial The degree of a polynomial is the degree of its term with largest degree. For example, x 2 + 2x is a binomial of degree 2, because the degree of its first term is 2 and the degree of its other term is less than 2.

17. 17 Find the degree of a polynomial 3x 3 y 2 + 4x 4 y 4 – 3x 3 is a trinomial of degree 8, because the degree of its second term is 8 and the degree of each of its other terms is less than 8. 25x 4 y 3 z 7 – 15xy 8 z 10 – 32x 8 y 8 z 3 + 4 is a polynomial of degree 19, because its second and third terms are of degree 19. Its other terms have degrees less than 19.

18. 18 Example Find the degree of each polynomial. a. –4x 3 – 5x 2 + 3x b. 5x 4 y 2 + 7xy 2 – 16x 3 y 5 c. –17a 2 b 3 c 4 + 12a 3 b 4 c 3, the degree of the first term because it has largest degree 8, the degree of the last term 9, the degree of the first term

19. 19 Find the degree of a polynomial If the polynomial contains a single variable, we usually write it with its exponents in descending order where the term with the highest degree is listed first, followed by the term with the next highest degree, and so on. If we reverse the order, the polynomial is said to be written with its exponents in ascending order.

20. 20 Evaluate a polynomial 4.

21. 21 Evaluate a polynomial When a number is substituted for the variable in a polynomial, the polynomial takes on a numerical value. Finding that value is called evaluating the polynomial.

22. 22 Example 4 Evaluate the polynomial 3x 2 + 2 when a. x = 0 b. x = 2 c. x = –3 d. x =.

23. 23 Example 4 – Solution a. 3x 2 + 2 = 3(0) 2 + 2 = 3(0) + 2 = 0 + 2 = 2 b. 3x 2 + 2 = 3(2) 2 + 2 = 3(4) + 2 = 12 + 2 = 14

24. 24 Example 4 – Solution c. 3x 2 + 2 = 3(–3) 2 + 2 = 3(9) + 2 = 27 + 2 = 29 d. cont’d

25. 25 Evaluate a polynomial function 5.

26. 26 Evaluate a polynomial function Since the right sides of the functions f (x) = 2x – 3, f (x) = x 2, and f (t) = –16t 2 + 64t are polynomials, they are called polynomial functions. We can evaluate these functions at specific values of the variable by evaluating the polynomial on the right side.

27. 27 Example 6 Given f (x) = 2x – 3, find f (–2). Solution: To find f (–2), we substitute –2 for x and evaluate the function. f (x) = 2x – 3 f (–2) = 2(–2) – 3 = –4 – 3 = –7 Thus, f (–2) = –7

28. 28 Graph a linear, quadratic, and cubic polynomial function 6.

29. 29 Graph a linear, quadratic, and cubic polynomial function We can graph polynomial functions. We make a table of values, plot points, and draw the line or curve that passes through those points. In the next example, we graph the function f (x) = 2x – 3. Since its graph is a line, it is a linear function.

30. 30 Example 7 Graph: f (x) = 2x – 3 Solution: We substitute numbers for x, compute the corresponding values of f (x), and list the results in a table, as in Figure 4-1. Figure 4-1

31. 31 Example 7 – Solution We then plot the pairs (x, f (x)) and draw a line through the points, as shown in the figure. From the graph, we can see that x can be any value. This confirms that the domain is the set of real numbers. We also can see that f (x) can be any value. This confirms the range is also the set of real numbers. cont’d

32. 32 Graph a linear, quadratic, and cubic polynomial function The function f (x) = x 2, is referred to as the squaring function. Since the polynomial on the right side is of second degree, we call this function a quadratic function. The function f (x) = x 3, is referred to as the cubing function. Since the polynomial on the right side is of third degree, we call this function a cubic function.