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Interactive Study Guide for Students Chapter 7: Polynomial Functions Section 1: Polynomial Functions Polynomial Functions Examples A example of a ____________ in ______ variable would be r2 – 3r + State the degree of the leading 1 because it contains only one variable, r. A polynomial of degree coefficient of each polynomial in one variable. If it is not a n in one variable x is an expression of the form: polynomial in one variable, aoxn + a1xn-1 + … + an-2x2 + an-1x + an where ao, a1,…an represent explain why: 1. 7x4 + 5x2 + x -9 real numbers, ao is not zero, and n represent a nonnegative integer. 2. 8x2 + 3xy -2y2 example: 3x5 + 2x4 – 5x3 + x2 + 1 where n = 5, ao = 3, a1 = 2, a2 = 5, a3 = 1, a4 = 0, a5 = 1 The __________ of a ___________ in one variable is the greatest exponent of the variable. The _____________ ______________ is the coefficient of the term with the highest degree. A polynomial equation used to represent a function is called a ______________ _________________. 3. 7x6 -4x3 + 1 x 4. Find p(a2) if p(x)=x3+4x2-5x 5. Find q(a+1) -2q(a) if q(x)=x2+3x+4 A polynomial function: P(x) = aoxn + a1xn-1 + … + an-2x2 + an-1x + an where ao, a1,…an represent real numbers, ao is not zero, and n represent a nonnegative integer. For each graph: Describe the end behavior. Determine whether it represents an odddegree or an evendegree polynomial. State the number of real zeros (roots). Graphs of Polynomial Functions Draw the general shapes of the polynomial functions below. These graphs show the ___________ number of times the graph of each type of polynomial may intersect the x-axis. Tell the degree of each function. Constant = ___ Linear =__ Quadratic = __ Cubic = __ Quartic = __ Quintic = __ 6. The _____ ___________is the behavior of the graph as x approaches positive infinity (______) and negative infinity (_____). This is represented as __________________, and _______________ respectively. Degree: even Degree: odd Leading coefficient: positive Leading coefficient: positive End behavior: End behavior: ___________ ____________ ___________ ____________ ___________ ____________ ___________ ____________ 7. 8. Degree: even Degree: odd Leading coefficient: negative Leading coefficient: negative End behavior: End behavior: ___________ ____________ ___________ ____________ ___________ ____________ ___________ ____________ Chapter 7: Polynomial Functions Section 2: Graphing Polynomial Functions Graph Polynomial Functions To graph a ____________ ___________, use a _____________ _________________. Examples 1. Graph f(x)=x4+ x3-4x2-4x Notice the end behaviors, the roots or zeros, and use the trace button to find them. The ____________ __________ is used to find the roots. If f(a)<0 and f(b)>0, then there is one root in between a and b. How many real roots? 2. f(x)=x3-5x2+3x+2 Maximum and Minimum Points A graph has a ________________ _______________ point if no other nearby points have a greater y-coordinate. Use trace to identify the roots: Likewise, a point is a ____________ _______________ point if no other nearby points have a lesser y-coordinate. 3. f(x)=x3-3x2+5 These are often referred to as ___________ ___________. The graph of a polynomial function of degree n has at most n-1 turning points. Estimate the x-coord. of relative max. & min. occur: 4. The average fuel consumed between 1960 and 2000 is f(t)= .025t3-1.5t2+18.25t+654 Chapter 7: Polynomial Functions Section 3:Solv. Eq. using Quadratic Techniques Quadratic Form Examples An expression that is in _____________ _________ can be written as au2 + bu + c for any numbers a, b, and c, a≠0, where u is some expression in x. 4 2 2 2 Write in quadratic form, if possible: 1. x4 + 13x2 + 35 2 Example: x – 16x + 60 can be written (x ) – 16(x ) + 60 by letting u = x2. 2. 16x6 – 625 3. 12x8 – x2 + 10 4. x – 9x 1 / 2 + 8 Solve. Solve Equations Using Quadratic Form 5. x4 -13x2 +36 = 0 You can solve higher-degree polynomial equations that can be written using _____________ ________ or have an expression that contains a ____________ ___________. 6. x3 + 343 = 0 7. x2/3 – 6x1/3 + 5 = 0 8. x - 6 x = 7 Chapter 7: Polynomial Functions Section 4: The Remainder and Factor Theorems Synthetic Substitution Examples Synthetic ____________ (learned in Chapter 5) is a shorthand method of long division. It can also be used to find the value of a function. f(x) = 2x4 – 5x2 + 8x -7 1.find f(6) using synthetic substitution Method 1: Long Division Method 2: Synthetic Division 4a + 5__________ a-2 2_ 4 4a2 – 3a + 6 4a2 – 8a 4 -3 6 8 10 5 16 2. find f(6) using direct substitution 5a + 6 5a – 10 16 3. Divide f(x)=x4+x32 17x -20x+32 by (x-4). Compare the remainder of 16 to f(2): f(2) = 4(2)2 – 3(2) + 6 = 16 This illustrates the _______________ ___________________ which is: If a polynomial f(x) is divided by x-a, the remainder is the constant f(a). When synthetic division is used to evaluate a function, it is called _______________ _________________. It is a convenient way of finding the value of a function, especially when the degree of the polynomial is greater than 2. Factors of Polynomials From example #4, notice that when you divide a polynomial by one of its binomial factors, the quotient is called a __________ __________. Since the remainder is 0, this means that (x-4) is a factor of the polynomial. This illustrates the ____________ ____________ which is a special case of the Remainder Theorem. Factor Theorem: the binomial x-a is a factor of the polynomial f(x) if and only if f(a)=0. 4. Show that x+3 is a factor of x3+6x2-x-30. Then find the remaining factors of the polynomial. 5. The volume of a rectangular prism is v(x)=x3+3x2-36x+32. One factor is x-4, find the missing measures. Chapter 7: Polynomial Functions Section 5: Roots and Zeros Types of Roots Examples Remember that a zero of a function f(x) is any value c such that f(c) =0. When the function is graphed, the real zeros of the function are the x-intercepts of the graph. Solve each equation. State the number and types of roots. 1. x + 3 = 0 n Let f(x)= anx + … + a1x + a0 be a polynomial function, then c is a zero of the polynomial function f(x), x – c is a factor of the polynomial function f(x),and c is a root or solution of the polynomial equation f(x) = 0. 2. x2 – 8x + 16 = 0 3. x3 +2x = 0 In addition, if c is a real number, then (c, 0) is an intercept of the graph of f(x). When you solve a polynomial equation with degree greater than zero, it may have ______ or _____ real roots, or __ real roots (the roots are imaginary numbers). Since real numbers and imaginary number both belong to the set of complex numbers, then we have the _____________ _____________ of ______________ which is that every polynomial equation with degree greater than zero has at ______ ___ root in the set of complex numbers. Descartes’ Rule: The number of _______ ______ _______ of y=P(x) is number of changes of sign of coefficients or less by an even number. 4. x4 – 1 = 0 Find the number of positive and negative zeros: 5. p(x)=x5-6x4-3x3+7x2-8x+1 The number of _______ _______ ______ of y=P(x)is number of changes of sign of coeff. Of P(-x) or less by even number. Find all the zeros: Find Zeros 6. g(x)=x3+6x2+21x+26 What strategies can we use to find the zeros? The _____________ ______________ Theorem states that if an imaginary is a zero, then its conjugate is also a zero. In other words, if a+bi is a zero, the a-bi is also. Chapter 7: Polyniomial Functions Identify Rational Theorems 7. Write a polynomial function of least degree with integral coefficients whose zeros include 3 and 2-i. Section 6: Rational Zero Theorem Examples Sometimes it is not possible to test all the possible zeros of a function using synthetic substitution. Use the _____________ __________ ____________ which is: List all the possible rational zeros of each function: 1. f(x) = 2x3-11x2+12x+9 If f(x) = aoxn + a1xn-1 + … + an-2x2 + an-1x + an is a polynomial function with integral coefficients, an p is a rational number in simplest form q 2. f(x) = x3 - 9x2 – x + 105 and is a zero of y = f(x), then p is a factor of an and q is a factor of a0. Example: 3 Let f(x) = 2x3 + 3x2 – 17x + 12. If is a zero of f(x), then 3 is a factor 2 of 12 and 2 is a factor of 2. All possible zeros would be all the combinations of the factors of 12 divided by the factors of 2, or __, __, __, __, __, __, and __. 3. The volume of a rectangular solid is 675 cm3. The width is 4cm less than the height, and the length is 6cm more than the height. Find the dimensions. p 1 2 -24 Find Rational Zeros Once you have found all the ____________ rational zeros, _________ each number using synthetic substitution. 67 5 1 3 5 9 4. Find all the zeros of f(x)=2x4-13x3+23x2-52x+60. p q 2 Chapter 7: Polynomial Functions Arithmetic Operations -13 23 -52 60 Section 7: Operations on Functions Examples Operation Definition Examples f(x)=x2-3x+1 g(x)=4x+5 1. Find (f+g)x 2. Find (f-g)x f(x)=x2+5x-1 g(x)-3x-2 3. Find (fg)x 4. Find ( Composition of Functions Suppose f and g are functions such that the _________ of g is a subset of the _________ of f. Then the composition function f o g can be described by the equation: (f o g)(x) = ________________ Example using mappings: f(x)={(3,4),(2,3),(-5,0)} 5),(4,3),(0,2)} fog g(x)={(3,- f )x g f(x)={(7,8),(5,3),(9,8)} g(x)={(5,7),(3,5),(7,9)} Find f o g and g o f. 5. gof f(x)=x+3 g(x)=x2+x-1 6. Find (f o g)(x) and f)(x) Chapter 7: Polynomial Functions Find Inverses (g o Section 8: Inverse Functions and Relations Examples Remember that a relation is a set of ordered pairs. The ___________ _____________ is the set of ordered pairs obtained by ___________ the coordinates of each of the original ordered pairs. The domain becomes the _________ and the range becomes the ___________. Example: Q = {(1, 2), (3, 4), (5, 6)} {(2, 1), (4, 3), (6, 5)} ___________ relations. S= Q and S are 1. The vertices of a right triangle are (2, 1), (5, 1) and (2, -4). Graph the triangle, find the vertices of the inverse relation, and determine if the resulting triangle is also a right triangle. Can you find the equation of the line that the triangle reflects across to get the inverse? The ordered pairs of __________ _____________ are also related and can be defined as f(x) and f-1(x). Suppose f and f-1 are inverse functions, then f(a) = b iff f-1(b) = a. In simple terms, change the x and the y and solve for y. 2. Find the inverse of f(x) = x6 2 Inverses of Relations and Functions You can determine whether two functions are inverses by finding both of their _______________. If both are the ___________ function, then they are inverses. 3. Determine whether f(x)=5x+10 and 1 5 g(x)= x–2 are inverse functions. [f o g](x) = x and [g o f] (x) = x When the inverse of a ___________ is a _________, then the original function is said to be _____-__-_____. To determine if the inverse of a function is a function, you can use the horizontal line test. Chapter 7: Polynomial Functions Square Root Functions Section 9: Square Root Functions and Inequalities Examples If a function contains a square root of a variable, it is called a __________ _______ ____________. The inverse of a quadratic function is a square root function only if the _________ is restricted to _____-_________ numbers. Calculator activity: (use the 2nd 1. Graph y = 3x 4 . What is the domain? What is the range? x-intercept? y-intercept? buttons) 1. Set window to [-2,8] &[-4,6]. Graph y= x , y= x + 1 and y= x - 2. State domain and range for each function, describe similarities and differences. 2. Set window to [0,10][0,10]. Graph y= x , y= 2 x and y= 8 x . State domain and range for each function, describe similarities and differences. 3. Set window to [-10,10][-10,10].How would you write an equation that translates the parent graph to the left three units. Test your equation. What did you get? Square Root Inequalities 2. A lookout on a submarine is h feet above the surface of the water. The greatest distance d in miles that the lookout can see on a clear day is given by the square root of the quantity h multiplied by 3 . Graph the function, 2 state the domain and range. A square root inequality is and ___________ (<,<,>,>) involving __________ _________. Use what you know about graphing inequalities. 4. Graph y< 2x 6 5. Graph y> x 1 3. A ship is 3 miles from the submarine. How high should the periscope be to see it?