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All Textbook Solutions for Algebra and Trigonometry (MindTap Course List)

58E 59E 60E 61E 62E 63E 64E 65E 66E 67E 68E 69E 70E 71-74 Finding a Polynomial from a Graph Find the polynomial of the specified degree whose graph is shown. Degree 3 72E 73E 74E 75E DISCOVER: Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q are the same. P(x)=3x45x3+x23x+5 Q(x)=(((3x5)x+1)x3)x+5 Try to evaluate P(2) and Q(2) in your head, using the forms given. Which is easier? Now write the polynomial R(x)=x52x4+3x32x2+3x+4 in nested form, like the polynomial Q. Use the nested form to find R(3) in your head. Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value of a polynomial using synthetic division?If the polynomial function Px=anxn+an-1xn-1++a1x+a0 has integer coefficients, then the only numbers that could possibly be rational zeros of P are all of the form pq, where p is a factor of __________ and q is a factor of __________. The possible rational zeros of Px=6x3+5x2-19x-10 are _________________________.2E 3E 4E 5-10 Possible Rational Zeros List all possible rational zeros given by the Rational Zeros Theorem but dont check to see which actually are zeros. Px=x3-4x2+3 6E 7E 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E 33E 34E 35E 36E 37E 38E 39E 40E 41E 42E 43E 44E 45E 46E 47E 48E 49E 45-54 Real Zeros of a Polynomial Find all the real zeros of the polynomial. Use the Quadratic Formula if necessary, as in Example 3a. Px=x5-4x4-x3+10x2+2x-4 51E 52E 53E 54E 55E 56E 57E 58E 59E 55-62 Real Zeros of a Polynomial A Polynomial P is given. a Find all real zeros of P.b Sketch a graph of P. Px=-x4+10x2+8x-8 61E 62E 63E 64E 65E 66E 67E 68E 69E 70E 71E 72E 73E 74E 75E 76E 77E 78E 79E 80E 81E 82E 83E 84E 81-86 Zeros of a Polynomial Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes Rule of Signs, the Quadratic Formula, or other factoring techniques. Px=x5-7x4+9x3+23x2-50x+24 86E 87E 88E 87-90 Polynomials With No Rational Zeros Show that the polynomial does not have any rational zeros Px=3x3-x2-6x+12 90E 91E 92E 93E 94E 95E 96E 97E 98E 99E 100E 101E 102E 103E Volume of a Box A rectangular box with a volume of 22 ft3 has a square base as shown below. The diagonal of the box between a pair of opposite corners is 1 feet longer than each side of the base. a If the base has sides of length x feet, show that x6-2x5-x4+8=0 b Show that two different boxes satisfy the given conditions. Find the dimensions in each case, rounded to the nearest hundredth of a foot.Girth of a Box A box with a square base has length plus girth of 108 in. Girth is the distance around the box. What is the length of the box if its volume is 2200 in3?106E 107E 108E 109E 110E 1E 2E 3E 4E 5E 6E 7E 8E 7-18. Complete Factorization A polynomial P is given. a Find all zeros of P, real and complex. b Factor P completely. P(x)=x32x2+2x 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E 33E 34E 35E 36E 37E 38E 37-46 Finding a Polynomial with Specified Zeros Find a polynomial with integer coefficients that satisfies the given conditions. Q has degree 3 and zeros 3, 2i, and 2i.40E 41E 42E 43E 44E 37-46 Finding a Polynomial with Specified Zeros Find a polynomial with integer coefficients that satisfies the given conditions. T has degree 4, zeros i and 1+i, and constant term 12.46E 47E 48E 49E 50E 51E 52E 53E 54E 55E 56E 57E 58E 59E 47-64 Finding Complex Zeros Find all zeros of the polynomial. P(x)=x4x2+2x+2 61E 62E 63E 64E 65E 66E 67E 68E 69E 70E 71E 72E 73E 74E 75E 76E If the rational function y=r(x) has the vertical asymptote x=2, then as x2+, either y or y.If the rational function y=r(x) has the horizontal asymptote y=2, then y as x.3-6 The following questions are about the rational function r(x)=(x+1)(x2)(x+2)(x3) The function r has x-intercepts __________ and __________.4E 5E 6E 7E 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E 33E 34E 35E 36E 37E 38E 39E 40E 41E 42E 43E 44E 45E 46E 47E 48E 49E 50E 51E 52E 53E 54E 55E 56E 57E 58E 59E 60E 61E 62E 63E 64E 65E 66E 67E 63-68 Rational Functions with Holes Find the factors that are common in the numerator and the denominator. Then find the intercepts and asymptotes, and sketch a graph of the rational function. State the domain and range of the function. r(x)=x2+4x5x3+7x2+10x 69E 70E 71E 72E 73E 74E 75E 76E 77E 78E 79E 80E 81E 81-86 End Behavior Graph the rational function, and find all vertical asymptotes, x- and y-intercepts, and local extrema, correct to the nearest tenth. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. y=x43x3+x23x+3x23x 83E 84E 85E 86E 87E Drug Concentration After a certain drug is injected into a patient, the concentration c of the drug in the bloodstream is monitored. At time t0 in minutes since the injection the concentration in mg/L is given by c(t)=30tt2+2 a Draw a graph of drug concentration. b What eventually happens to the concentration of drug in the bloodstream?Drug Concentration A drug is administered to a patient, and the concentration of the drug in the bloodstream is monitored. At time t0 in hours since giving the drug the concentration in mg/L is given by c(t)=5tt2+1 Graph the function c with a graphing device. a What is the highest concentration of drug that is reached in the patients bloodstream? b What happens to the drug concentration after a long period of time? c How long does it take for the concentration to drop below 0.3 mg/L?90E 91E Focusing Distance For a camera with a lens of fixed focal length F to focus on an object located a distance x from the lens, the film must be placed a distance y behind the lens, where F, x, and y are related by 1x+1y=1F See the figure. Suppose the camera has a 55-mm lens F=55. a Express y as a function of x, and graph the function. b What happens to the focusing distance y as the object moves far away from the lens? c What happens to the focusing distance y as the object moves close to the lens?DISCUSS: Constructing a Rational Function from its Asymptotes Give an example of a rational function that has vertical asymptote x=3. Now give an example of one that has vertical asymptote x=3 and horizontal asymptote y=2. Now give an example of a rational function with vertical asymptotes x=1 and x=1, horizontal asymptote y=0, and x-intercept 4.DISCUSS: A Rational Function with No Asymptote Explain how you can tell without graphing it that the function r(x)=x6+10x4+8x2+15 has no x-intercept and no horizontal, vertical, or slant asymptote. What is its end behavior?95E

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