Any questions on the Section 5 2 homework Now please CLOSE YOUR LAPTOPS and turn off and put away your cell phones Sample Problems Page Link Dr Bruce Johnston 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 Consider the polynomial 7x5 x2y2 4xy 7 How many TERMS does it have There are 4 terms 7x5 x2y2 4xy and 7 What are the coefficients of those terms The coefficient of term 7x5 is 7 The coefficient of term x2y2 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 A Monomial is a polynomial with 1 term A Binomial is a polynomial with 2 terms A Trinomial is a polynomial with 3 terms 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 7x4 is 4 Degree of a constant like 9 is 0 because you could write it as 9x0 since x0 1 Degree of the term 5a4b3c is 8 add all of the exponents on all variables remembering that c can be written as c1 Degree of a polynomial To find the degree take the largest degree of any term of the polynomial Example The degree of 9x3 4x2 7 is 3 More examples 1 Consider the polynomial 7x5 x3y3 4xy Is it a monomial binomial or trinomial What is the degree of the polynomial 2 Which of the following expressions are NOT polynomials 5x4 5x 5x 3y7 2xy 10 1 x 5 3x 5 5x3y7 2xy 10 3 y2 6y 8 Problem from today s homework We can use function notation to represent polynomials Example P x 2x3 3x 4 is a polynomial function Evaluating a polynomial for a particular value involves replacing the value for the variable s involved Example Find the value P 2 2x3 3x 4 P 2 2 2 3 3 2 4 2 8 6 4 This means that the ordered pair 6 2 6 would be one point on the graph of this function Don t forget how to work with fractions Example For the polynomial function f x 7x2 x 2 Calculate f Answer Calculate f Answer 14 9 Like terms Terms that contain exactly the same variables raised to exactly the same powers Warning Only like terms can be combined by combining their coefficients Example Combine like terms to simplify x2y xy y 10x2y 2y xy x2y 10x2y xy xy y 2y like terms are grouped together 1 10 x2y 1 1 xy 1 2 y 11x2y 2xy 3y 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 Example Add or subtract each of the following as indicated 1 3x 8 4x2 3x 3 3x 8 4x2 3x 3 4x2 3x 3x 8 3 4x2 5 2 4 y 4 4 y 4 y 4 4 y 8 3 a2 1 a2 3 5a2 6a 7 a2 1 a2 3 5a2 6a 7 a2 a2 5a2 6a 1 3 7 3a2 6a 11 Problem from today s homework Problem from today s homework Application Problems 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 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 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 Graphing Polynomial Functions 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 ax2 bx c the graph is a parabola opening up when a 0 and opening down when a 0 a 0 a 0 x x Examples related to today s homework 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 Graph P x x2 Graph P x x2 5 Graph P x 3x2 Graph P x 3x2 Graph P x 3x2 1 Graph P x 3x2 2x 1 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 Problem from today s homework 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 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