1. Any questions on the Section 5.2 homework?

2. Please CLOSE YOUR LAPTOPS, and turn off and put away your cell phones, and get out your note- taking materials.

3. Section 5.3 Polynomials and Polynomial Functions Polynomial vocabulary: Term – a number or a product of a number and variables raised to powers (the terms in a polynomial are separated by + or - signs) Coefficient – numerical factor of a term Constant – term which is only a number A polynomial is a sum of terms involving coefficients (numbers) times variables raised to a whole number (0, 1, 2, …) exponent, with no variables appearing in any denominator.

4. Consider the polynomial 7x 5 + x 2 y 2 – 4xy + 7 How many TERMS does it have? There are 4 terms: 7x 5, x 2 y 2, -4xy and 7. What are the coefficients of those terms? The coefficient of term 7x 5 is 7, The coefficient of term x 2 y 2 is 1, The coefficient of term –4xy is –4 The coefficient of term 7 is 7. 7 is a constant term. (no variable part, like x or y)

5. A Monomial is a polynomial with 1 term. A Binomial is a polynomial with 2 terms. A Trinomial is a polynomial with 3 terms.

6. Degree of a term: To find the degree, take the sum of the exponents on the variables contained in the term. Degree of the term 7x 4 is 4 Degree of a constant (like 9) is 0. (because you could write it as 9x 0, since x 0 = 1) Degree of the term 5a 4 b 3 c is 8 (add all of the exponents on all variables, remembering that c can be written as c 1 ). Degree of a polynomial: To find the degree, take the largest degree of any term of the polynomial. Example: The degree of 9x 3 – 4x 2 + 7 is 3.

7. More examples: 1. Consider the polynomial 7x 5 + x 3 y 3 – 4xy Is it a monomial, binomial or trinomial? What is the degree of the polynomial? 2. Which of the following expressions are NOT polynomials? _ 5x 4 - √5x + Π  -5x -3 y 7 + 2xy – 10 1  3x + 5 x + 5 -5x 3 y 7 + 2xy – 10  y 2 + 6y - 8 3

8. Problem from today’s homework:

9. We can use function notation to represent polynomials. Example: P(x) = 2x 3 – 3x + 4 is a polynomial function. Evaluating a polynomial for a particular value involves replacing the value for the variable(s) involved. Find the value P(-2) = 2x 3 – 3x + 4. Example = 2(-2) 3 – 3(-2) + 4P(-2) = 2(-8) + 6 + 4 = -6 This means that the ordered pair (-2, -6) would be one point on the graph of this function.

10. Don’t forget how to work with fractions! Example: For the polynomial function f(x) = 7x 2 + x – 2 Calculate f(½) (Answer: ¼) Calculate f(-⅓) (Answer: 14 ) 9

11. Like terms Terms that contain exactly the same variables raised to exactly the same powers. Combine like terms to simplify. x 2 y + xy – y + 10x 2 y – 2y + xy = Only like terms can be combined by combining their coefficients. Warning! Example 11x 2 y + 2xy – 3y(1 + 10)x 2 y + (1 + 1)xy + (-1 – 2)y = x 2 y + 10x 2 y + xy + xy – y – 2y = (like terms are grouped together)

12. Adding polynomials Combine all the like terms. Subtracting polynomials Change the signs of the terms of the polynomial being subtracted, and then combine all the like terms.

13. 3a 2 – 6a + 11 Example Add or subtract each of the following, as indicated. 1) (3x – 8) + (4x 2 – 3x +3) = 4x 2 + 3x – 3x – 8 + 3 = 4x 2 – 5 2) 4 – (-y – 4) = 4 + y + 4 = y + 4 + 4= y + 8 3) (-a 2 + 1) – (a 2 – 3) + (5a 2 – 6a + 7) = -a 2 + 1 – a 2 + 3 + 5a 2 – 6a + 7 = -a 2 – a 2 + 5a 2 – 6a + 1 + 3 + 7 = = 3x – 8 + 4x 2 – 3x + 3

14. Problem from today’s homework:

16. In the previous chapter, we examined Cost and Revenue functions. A Profit function for businesses can be found by using Revenue – Cost. This is denoted P(x) = R(x) – C(x). Application Problems:

17. Example Baskets, Inc., is planning to introduce a new woven basket. The company estimates that $640 worth of new equipment will be needed to manufacture this new type of basket and that it will cost $15 per basket to manufacture. The company also estimates that the revenue from each basket will be $31. Find the profit function. Solution: R(x) = 31x and C(x) = 15x + 640. So P(x) = R(x) – C(x) = 31x – (15x + 640) = 16x – 640

18. Example (ct’d) Now use this function to calculate the profit that will be earned if a total of 110 baskets are produced: Solution: Previously we showed that P(x) = 16x – 640, So now just plug 110 in for x: P(110) = 16·110 – 640 = 1760 – 640 = 1120 ANSWER: The profit on 110 baskets will be $1120.

19. Using the degree of a polynomial, we can determine what the general shape of the function will be, before we ever graph the function. A polynomial function of degree 1 is a linear function. We have examined the graphs of linear functions in great detail previously in this course and prior courses. A polynomial function of degree 2 is a quadratic function. We briefly examined graphs of quadratics in Chapter 3. In general, for the quadratic equation of the form y = ax 2 + bx + c, the graph is a parabola opening up when a > 0, and opening down when a < 0. a > 0a < 0 xx Graphing Polynomial Functions:

20. Examples related to today’s homework: Graph P(x) = x 2 Graph P(x) = x 2 – 5 Graph P(x) = 3x 2 Graph P(x) = -3x 2 Graph P(x) = -3x 2 + 1 Graph P(x) = -3x 2 +2x + 1 To help you identify the graph of each of these quadratic polynomials, start by answering these questions: Does the parabola open upward or downward? What is the y-intercept of the graph? Note: Remember that if you need to graph the function completely (i.e. for a problem that doesn’t just ask you to chose the correct graph from a list), you would need to calculate at least 5 or 6 ordered pairs and plot them on an x-y coordinate system.

21. Problem from today’s homework:

22. Reminder: This homework assignment on Section 5.3 is due at the start of next class period. You’re always welcome to stay and work on your homework in the open lab next door after class.

23. Math TLC Open Lab Hours: Next door in room 203 Monday - Thursday 8:00 a.m. – 6:30 p.m. Teachers and tutors available for one-on-one help on homework and practice quiz/test problems. NO APPOINTMENTS NECESSARY – JUST DROP IN AT EITHER PLACE.