Chapter 3PolynomialsSection 3.1: An Introduction to Polynomial FunctionsPolynomials and Polynomial FunctionsAnswer the following.Section 3.2: Adding, Subtracting, and MultiplyingPolynomialsOperations with PolynomialsMultiply. Write your final answer with the terms in descending order, from highest to lowest degree.Section 3.3: Dividing PolynomialsPolynomial Long Division and Synthetic DivisionUse long division to find the quotient and the remainder.Section 3.4: Quadratic FunctionsThe Definition and Graph of a Quadratic Function(c) Find the y-intercept of the parabola.SECTION 3.1 An Introduction to Polynomial FunctionsChapter 3 PolynomialsSection 3.1: An Introduction to Polynomial Functions Polynomials and Polynomial FunctionsPolynomials and Polynomial Functions Polynomials: A polynomial in a single variable x is the sum of a finite number of terms of the formnax, where a is a constant and the exponent n is a whole number. Recall that the setof whole numbers is { }0,1, 2, ...Examples of polynomials in x are 33 , 5 8 ,x x x+ and 24 7 1x x- +. They can beclassified according to the number of terms:MATH 1300 Fundamentals of Mathematics203CHAPTER 3 PolynomialsUniversity of Houston Department of Mathematics204SECTION 3.1 An Introduction to Polynomial FunctionsDegree of a Polynomial: Example: MATH 1300 Fundamentals of Mathematics205CHAPTER 3 PolynomialsSolution: University of Houston Department of Mathematics206SECTION 3.1 An Introduction to Polynomial FunctionsPolynomial Functions: Evaluating Polynomial Functions: MATH 1300 Fundamentals of Mathematics207CHAPTER 3 PolynomialsExample: Solution: Graphs of Polynomial Functions: University of Houston Department of Mathematics208SECTION 3.1 An Introduction to Polynomial FunctionsExample: Solution: MATH 1300 Fundamentals of Mathematics209CHAPTER 3 PolynomialsAdditional Example 1: Solution: University of Houston Department of Mathematics210SECTION 3.1 An Introduction to Polynomial FunctionsAdditional Example 2: Solution: Leading term:Degree:Leading Coefficient:Constant term:MATH 1300 Fundamentals of Mathematics211CHAPTER 3 PolynomialsAdditional Example 3: Solution: Additional Example 4: University of Houston Department of Mathematics212SECTION 3.1 An Introduction to Polynomial FunctionsSolution: Additional Example 5: Solution: MATH 1300 Fundamentals of Mathematics213CHAPTER 3 PolynomialsUniversity of Houston Department of Mathematics214Exercise Set 3.1: An Introduction to Polynomial FunctionsAnswer the following.(a) State whether or not each of the following expressions is a polynomial. (Yes or No.) (b) If the answer to part (a) is yes, then state the degree of the polynomial.(c) If the answer to part (a) is yes, then classify the polynomial as a monomial, binomial, trinomial, or none of these. (Polynomials of four or more terms are not generally given specific names.) 1.34 3x+2.5 386 3x xx+ +3.3 5x -4.3 22 4 7 4x x x+ - -5.3 225 6 74 5x xx x- +- +6. 87.3 25 8x x x- -- +8.27 52 39x x- +9.3 27 5 32xx x+ - -10.1 4 13 7 2x x- -- +11.11 134 29 2 4x x x- + -12.23 1x x- +13.632x-14.3 26 8x xx+15.43 7x- +16.2 110 3 5x x- -+ +17.10-18.7 4x-19.1 5 1 3 1 25 2 3 4x x x- - -+ + +20.2 4 93 5 6 3x x x- + + -21.3 4 2 23 2a b a b-22.5 2 4 94 3x y x y- -- -23.5 3234x yxy+24.2 9 3 4 22 15 43x y z xy x y z+ -25.3 7 4 3 2325 74xyz y x y z- - -26.7 3 5 6 2 42 3a a b b a b- + + -Answer True or False. 27. (a)37 2x x- is a trinomial.(b)37 2x x- is a third degree polynomial.(c)37 2x x- is a binomial.(d)37 2x x- is a first degree polynomial28. (a)25 3 2x x- + is a trinomial.(b)25 3 2x x- + is a third degree polynomial.(c)25 3 2x x- + is a binomial.(d)25 3 2x x- + is a second degree polynomial.29. (a)36x- is a monomial.(b)36x- is a third degree polynomial.(c)36x- is a first degree polynomial.(d)36x- is a trinomial.30. (a)2 34 7x x x- + is a second degree polynomial.(b)2 34 7x x x- + is a binomial.(c)2 34 7x x x- + is a third degree polynomial.(d)2 34 7x x x- + is a trinomial.31. (a)7 4 6 83 2 3x x y y- - is a tenth degree polynomial.(b)7 4 6 83 2 3x x y y- - is a binomial.(c)7 4 6 83 2 3x x y y- - is an eighth degree polynomial.(d)7 4 6 83 2 3x x y y- - is a trinomial.32. (a)4 53a b- is a fifth degree polynomial.(b)4 53a b- is a trinomial.(c)4 53a b- is a ninth degree polynomial.(d)4 53a b- is a monomial.MATH 1300 Fundamentals of Mathematics215Exercise Set 3.1: An Introduction to Polynomial FunctionsEach of the graphs below represents a polynomial function. Use the graph to determine the x- and y-intercept(s). The equation of each graph is given for informational purposes, but the intercepts can be determined entirely from the graph.33.3 2( ) 5 2 8f x x x x= - + +34.2( ) 8 12f x x x=- + -35.5 4 3( ) 2 16 36 54f x x x x x= - + -36.4 3 27 5 3113 3 3 3( ) 10f x x x x x=- - - + +University of Houston Department of Mathematics216         xy                 xy          xy          xyExercise Set 3.1: An Introduction to Polynomial FunctionsFor each of the following polynomial functions,(a) Classify the function as linear, quadratic, or cubic.(b) Find the x- and y-intercept(s) of the function. (Do this algebraically without drawing the graph.)(c) Find ( 4), ( 1)f f- - and (6)f.37.2( ) 64f x x= -38.( ) 3 8f x x= +39.3( ) 32 4f x x= -40.2( ) 50 2f x x= -41.( ) 12 5f x x= -42.3( ) 2 54f x x= -Follow the directions above for numbers 43 and 44, but in part (b), find the y-intercept only. (Do not find the x-intercept(s).)43.2( ) 3 28f x x x= - -44.2( ) 18 9 2f x x x= + -MATH 1300 Fundamentals of Mathematics217CHAPTER 3 PolynomialsSection 3.2: Adding, Subtracting, and MultiplyingPolynomials Operations with PolynomialsOperations with Polynomials Like Terms: Addition of Polynomials: Example: University of Houston Department of Mathematics218SECTION 3.2 Adding, Subtracting, and Multiplying PolynomialsSolution: Subtraction of Polynomials: Example: Solution: Multiplication of Polynomials: MATH 1300 Fundamentals of Mathematics219CHAPTER 3 PolynomialsExample: Solution: Special Case - Multiplying Two Binomials: Example: University of Houston Department of Mathematics220SECTION 3.2 Adding, Subtracting, and Multiplying PolynomialsSolution: Special Products: Example: MATH 1300 Fundamentals of Mathematics221CHAPTER 3