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All Textbook Solutions for Precalculus

78PE 79PE 80PE 81PE Explain how to use the discriminant to determine the number of x-intercepts for the graph of fx=ax2+bx+c.83PE 84PE For Exercises 85-88, define a quadratic function y=fx that satisfies the given conditions. Vertex 2,3 and passes through 0,5 For Exercises 85-88, define a quadratic function y=fx that satisfies the given conditions. Vertex 3,1 and passes through 0,17 For Exercises 85-88, define a quadratic function y=fx that satisfies the given conditions. Axis of symmetry x=4, maximum value 6, passes through 1,3 For Exercises 85-88, define a quadratic function y=fx that satisfies the given conditions. Axis of symmetry x=2, minimum value 5, passes through 2,13 For Exercises 89-92, find the value of b or c that gives the given minimum or maximum value. fx=2x2+12x+c;minimumvalue9 For Exercises 89-92, find the value of b or c that gives the given minimum or maximum value. fx=3x2+12x+c;minimumvalue4 91PE For Exercises 89-92, find the value of b or c that gives the given minimum or maximum value. fx=x2+bx2;maximumvalue7 Use the leading term to determine the end behaviour of the graph of the function. a.fx=0.3x45x23x+4b.gx=67x94x+423x5 Find the zeros of the function defined by fx=4x34x225x+25.Find the zeros of the function defined by fx=x3+10x2+25x.Determine the zeros and their multiplicities for the given functions. a.px=35x+345x15b.qx=2x614x4 Show that fx=x4+6x326x+15 has a zero on the interval 4,3.Graph gx=x3+4x.Graph hx=0.5xx1x+32.A function defined by fx=anxn+an1xn1+an2xn2++a1x+a0wherean,an1,an2,,a1,a0 are real numbers and an0 is called a function.The function given by fx=3x5+2x+12x (is/is not) a polynomial function.The function given by fx=3x5+2x+2x (is/is not) a polynomial function.A quadratic function is a polynomial function of degree .A linear function is a polynomial function of degree .The values of x in the domain of a polynomial function f for which fx=0 are called the of the function.What is the maximum number of turning points of the graph of fx=3x64x55x4+2x2+6?If the graph of a polynomial function has 3 turning points, what is the minimum degree of the function?If c is a real zero of a polynomial function and the multiplicity is 3, does the graph of the function cross the x-axis or touch the x-axis (without crossing) at c,0?If c is a real zero of a polynomial function and the multiplicity is 6, does the graph of the function cross the x-axis or touch the x-axis (without crossing) at c,0?11PE What is the leading term of fx=13x343x+52?13PE For Exercises 13-20, determine the end behavior of the graph of the function. (See Example 1) gx=12x6+8x4x3+9 For Exercises 13-20, determine the end behavior of the graph of the function. (See Example 1) hx=12x5+8x44x38x+1 For Exercises 13-20, determine the end behavior of the graph of the function. (See Example 1) kx=11x74x2+9x+3 17PE For Exercises 13-20, determine the end behavior of the graph of the function. (See Example 1) nx=2x+43x13x+5 For Exercises 13-20, determine the end behavior of the graph of the function. (See Example 1) px=2x23x2x33 For Exercises 13-20, determine the end behavior of the graph of the function. (See Example 1) qx=5x42x32x+5 Given the function defined by gx=3x13x+54, the value 1 is a zero with multiplicity and the value 5 is a zero with multiplicity .Given the function defined by hx=12x5x+0.63, the value 0 is a zero with multiplicity and the value 0.6 is a zero with multiplicity .For Exercises 23-38, find the zeros of the function and state the multiplicities. (See Example 2-4) fx=x3+2x225x50 For Exercises 23-38, find the zeros of the function and state the multiplicities. (See Example 2-4) gx=x3+5x2x5 For Exercises 23-38, find the zeros of the function and state the multiplicities. (See Example 2-4) hx=6x39x2+60x For Exercises 23-38, find the zeros of the function and state the multiplicities. (See Example 2-4) kx=6x3+26x228x For Exercises 23-38, find the zeros of the function and state the multiplicities. (See Example 2-4) mx=x510x4+25x3 28PE For Exercises 23-38, find the zeros of the function and state the multiplicities. (See Example 2-4) px=3xx+23x+4 For Exercises 23-38, find the zeros of the function and state the multiplicities. (See Example 2-4) qx=2x4x+13x22 For Exercises 23-38, find the zeros of the function and state the multiplicities. (See Example 2-4) tx=5x3x52x+9x3x+3 32PE For Exercises 23-38, find the zeros of the function and state the multiplicities. (See Example 2-4) cx=x35x3+5 For Exercises 23-38, find the zeros of the function and state the multiplicities. (See Example 2-4) dx=x211x2+11 For Exercises 23-38, find the zeros of the function and state the multiplicities. (See Example 2-4) fx=4x437x2+9 36PE For Exercises 23-38, find the zeros of the function and state the multiplicities. (See Example 2-4) nx=x67x4 For Exercises 23-38, find the zeros of the function and state the multiplicities. (See Example 2-4) mx=x55x3 39PE For Exercises 39-40, determine whether the intermediate value theorem guarantees that the function has a zero on the given interval. (See Example 5) gx=2x313x2+18x+5a.1,2b.2,3c.3,4d.4,5 For Exercises 41-42, a table of values is given for Y1=fx. Determine whether the intermediate value theorem guarantees that the function has a zero on the given interval. Y1=21x4+46x3238x2506x+77 a.4,3b.3,2c.2,1d.1,0 For Exercises 41-42, a table of values is given for Y1=fx. Determine whether the intermediate value theorem guarantees that the function has a zero on the given interval. Y1=10x4+21x3119x2147x+343 a.4,3b.3,2c.2,1d.1,0 Given fx=4x38x225x+50, a. Determine if f has a zero on the interval 3,2. b. Find a zero of f on the interval 3,2.44PE For Exercises 45-52, determine if the graph can represent a polynomial function. If so. Assume that the end behaviour and all turning points are represented in the graph. a. Determine the minimum degree of the polynomial. b. Determine whether the leading coefficient is positive or negative based on the end behaviour and whether the degree of the polynomial is odd or even. c. Approximate the real zeros of the function, and determine if their multiplicities are even or odd.46PE For Exercises 45-52, determine if the graph can represent a polynomial function. If so. Assume that the end behaviour and all turning points are represented in the graph. a. Determine the minimum degree of the polynomial. b. Determine whether the leading coefficient is positive or negative based on the end behaviour and whether the degree of the polynomial is odd or even. c. Approximate the real zeros of the function, and determine if their multiplicities are even or odd.48PE For Exercises 45-52, determine if the graph can represent a polynomial function. If so. Assume that the end behaviour and all turning points are represented in the graph. a. Determine the minimum degree of the polynomial. b. Determine whether the leading coefficient is positive or negative based on the end behaviour and whether the degree of the polynomial is odd or even. c. Approximate the real zeros of the function, and determine if their multiplicities are even or odd.50PE For Exercises 45-52, determine if the graph can represent a polynomial function. If so. Assume that the end behaviour and all turning points are represented in the graph. a. Determine the minimum degree of the polynomial. b. Determine whether the leading coefficient is positive or negative based on the end behaviour and whether the degree of the polynomial is odd or even. c. Approximate the real zeros of the function, and determine if their multiplicities are even or odd.For Exercises 45-52, determine if the graph can represent a polynomial function. If so. Assume that the end behaviour and all turning points are represented in the graph. a. Determine the minimum degree of the polynomial. b. Determine whether the leading coefficient is positive or negative based on the end behaviour and whether the degree of the polynomial is odd or even. c. Approximate the real zeros of the function, and determine if their multiplicities are even or odd.53PE For Exercises 53-58, a. Identify the power function of the form y=xn that is the parent function to the given graph. b. In order, outline the transformations that would be required on the graph of y=xn to make the graph of the given function. See Section 1.6 page 190. c. Match the function with the graph of I-VI. fx=12x34 For Exercises 53-58, a. Identify the power function of the form y=xn that is the parent function to the given graph. b. In order, outline the transformations that would be required on the graph of y=xn to make the graph of the given function. See Section 1.6 page 190. c. Match the function with the graph of I-VI. kx=x+23+3 56PE For Exercises 53-58, a. Identify the power function of the form y=xn that is the parent function to the given graph. b. In order, outline the transformations that would be required on the graph of y=xn to make the graph of the given function. See Section 1.6 page 190. c. Match the function with the graph of I-VI. mx=x35+1 58PE For Exercises 59-76, sketch the function. (See Example 6-7) fx=x35x2 For Exercises 59-76, sketch the function. (See Example 6-7) gx=x52x4 61PE For Exercises 59-76, sketch the function. (See Example 6-7) hx=14x1x4x+2 For Exercises 59-76, sketch the function. (See Example 6-7) kx=x4+2x38x2 For Exercises 59-76, sketch the function. (See Example 6-7) hx=x4x36x2 For Exercises 59-76, sketch the function. (See Example 6-7) kx=0.2x+22x43 For Exercises 59-76, sketch the function. (See Example 6-7) mx=0.1x32x+13 67PE 68PE For Exercises 59-76, sketch the function. (See Example 6-7) tx=x4+11x228 70PE For Exercises 59-76, sketch the function. (See Example 6-7) gx=x4+5x24 For Exercises 59-76, sketch the function. (See Example 6-7) hx=x4+10x29 For Exercises 59-76, sketch the function. (See Example 6-7) cx=0.1xx24x+23 74PE For Exercises 59-76, sketch the function. (See Example 6-7) mx=110x+3x3x+13 For Exercises 59-76, sketch the function. (See Example 6-7) fx=110x1x+3x42 For Exercises 77-88, determine if the statement is true or false. If a statement is false, explain why. The function defined by fx=x+15x52 crosses the x-axis at 5.For Exercises 77-88, determine if the statement is true or false. If a statement is false, explain why. The function defined by gx=3x+42x34 touches but does not cross the x-axis at 32,0.For Exercises 77-88, determine if the statement is true or false. If a statement is false, explain why. A third-degree polynomial has three turning points.For Exercises 77-88, determine if the statement is true or false. If a statement is false, explain why. A third-degree polynomial has two turning points.For Exercises 77-88, determine if the statement is true or false. If a statement is false, explain why. There is more than one polynomial function with zeros of 1, 2, and 6.For Exercises 77-88, determine if the statement is true or false. If a statement is false, explain why. There is exactly one polynomial with integer coefficients with zeros of 2, 4, and 6.For Exercises 77-88, determine if the statement is true or false. If a statement is false, explain why. The graph of a polynomial function with leading term of even degree is up to the far left and up to the far right.For Exercises 77-88, determine if the statement is true or false. If a statement is false, explain why. If c is a real zero of an even polynomial function, then c is also a zero of the function.For Exercises 77-88, determine if the statement is true or false. If a statement is false, explain why. The graph of fx=x327 has three x-intercepts.For Exercises 77-88, determine if the statement is true or false. If a statement is false, explain why. The graph of fx=3x2x44 has no points in Quadrants III or IV.For Exercises 77-88, determine if the statement is true or false. If a statement is false, explain why. The graph of px=5x4x+12 has no points in Quadrants I or II.For Exercises 77-88, determine if the statement is true or false. If a statement is false, explain why. A fourth-degree polynomial has exactly two relative minima and two relative maxima.A rocket will carry a communications satellite into low Earth orbit. Suppose that the thrust during the first 200 sec of flight is provided by solid rocket boosters at different points during liftoff. The graph shows the acceleration in G-forces (that is, acceleration in 9.8-m/sec2 increments) versus time after launch. a. Approximate the interval(s) over which the acceleration is increasing. b. Approximate the interval(s) over with the acceleration is decreasing. c. How many turning points does the graph show? d. Based on the number of turning points, what is the minimum degree of a polynomial function that could be used to model acceleration versus time? Would the leading coefficient be positive or negative? e. Approximate the time when the acceleration was the greatest. f. Approximate the value of the maximum acceleration.Data from a 20-yr study show the number of new AIDS cases diagnosed among 20- to 24-yr-olds in the United States x years after the study began. a. Approximate the interval(s) over which the number of new AIDS cases among 20- to 24-yr-olds increased. b. Approximate the interval(s) over which the number of new AIDS cases among 20- to 24-yr-olds decreased. c. How many turning points does the graph show? d. Based on the number of turning points, what is the minimum degree of a polynomial function that could be used to model the data? Would the leading coefficient be positive or negative? e. How many years after the study began was the number of new AIDS cases among 20- to 24-yr-olds the greatest? f. What was the maximum number of new cases diagnosed in a single year?Given a polynomial function defined by y=fx, explain how to find the x-intercepts.Given a polynomial function, explain how to determine whether an x-intercept is a touch point or a cross point.93PE 94PE Use this broader statement of the intermediate value theorem for Exercises 95-96. Given fx=x23x+2, a. Evaluate f3andf4. b. Use the intermediate value theorem to show that there exists at least one value of x for which fx=4 on the interval 3,4. c Find the value(s) of x for which fx=4 on the interval 3,4.Use this broader statement of the intermediate value theorem for Exercises 95-96. Given fx=x24x+3, a. Evaluate f4andf3. b. Use the intermediate value theorem to show that there exists at least one value of x for which fx=5 on the interval 4,3. c. Find the value(s) of x for which fx=5 on the interval 4,3.For a certain individual the volume (in liters) of au in the lungs during a 4.5-sec respiratory cycle is shown in the table for 0.5-sec intervals. Graph the points and then find a third-degree polynomial function to model the volume Vt for t between 0 sec and 4.5 sec.The torque (in ft-lb) produced a certain automobile engine turning at x thousand revolutions per minute is shown in the table. Graph the points and then find a third-degree polynomial to model the torque Txfor1x5.A solar oven is to be made from an open box with reflective sides. Each box is made from a 30-in. by 24-in. rectangular sheet of aluminium with square of length x (in inches) removed from each corner. Then the flaps are folder up to form an open box. a. Show that the volume of the box is given by Vx=4x3108x2+720xfor0x12. b. Graph the function from part (a) and a â€œmaximumâ€� feature on a graphing utility to approximate the length of the sides of the squares that should be removed to maximize the volume. Round to the nearest tenth of an inch. c. Approximate the maximum volume. Round to the nearest cubic inch.For Exercises 100-101, two viewing windows are given for the graph of y=fx. Choose the window that best shows the key features of the graph. fx=2x0.5x0.1x+0.2a.10,10,1by10,10,1b.1,1,0.1by0.05,0.05,0.01 For Exercises 100-101, two viewing windows are given for the graph of y=fx. Choose the window that best shows the key features of the graph. gx=0.08x16x+2x3a.10,10,1by10,10,1b.5,20,5by50,30,10 For Exercises 102-103, graph the function defined by y=fx on an appropriate viewing window. kx=1100x20x+1x+8x6 For Exercises 102-103, graph the function defined by y=fx on an appropriate viewing window. px=x0.4x+0.5x+0.1x0.8 Use long division to divide 4x323x+32x5.Use long division to divide. 17x+5x23x3+2x4x2+3 Use long division to divide. 3x214x+15x3 Use synthetic division to divide. 4x328x7x3 Use synthetic division to divide. 3x+7x3+5+2x4x+1 Given fx=x4+x36x25x15, use the remainder theorem to evaluate a.f5b.f3 Use the remainder theorem to determine if the given number, c, is a zero of the function. a.fx=2x43x2+5x11;c=2b.fx=2x3+5x214x35;c=7c.fx=x37x2+16x10;c=3+i Use the factor theorem to determine if the given polynomials are factors of fx=2x413x3+10x225x+6. a.x6b.x+3 a. Factor fx=2x3+7x214x40, given that 4 is a zero of f . b. Solve the equation. 2x3+7x214x40=0 Write a polynomial fx of degree 3 that has the zeros 13,3,and3.Given the division algorithm, identify the polynomials representing the dividend, divisor, quotient, and reminder. fx=dxqx+rx Given 2x35x26x+1x3=2x2+x3+8x3, use the division algorithm to check the result.The remainder theorem indicates that if a polynomial fx is divided by xc, then the remainder is .Given a polynomial fx, the factor theorem indicates that if fc=0,thenxc is a of fx, furthermore, if xc is a factor of fx,thenfc=.Answer true or false. If 5 is a zero of a polynomial, then x5 is a factor of the polynomial.Answer true or false. If x+3 is a factor of a polynomial, then 3 is a zero of the polynomial.For Exercises 7-8, (See Example 1) a. Use long division to divide. b. Identify the dividend, divisor, quotient, and remainder. c. Check the result from part (a) with the division algorithm. 6x2+9x+52x5 8PE For Exercises 9-22, use long division to divide. (See Example 1-3) 3x311x210x4 For Exercises 9-22, use long division to divide. (See Example 1-3) 2x37x265x5 For Exercises 9-22, use long division to divide. (See Example 1-3) 8+30x27x212x3+4x4x+2 For Exercises 9-22, use long division to divide. (See Example 1-3) 4828x+20x2+17x3+3x4x+3 For Exercises 9-22, use long division to divide. (See Example 1-3) 20x2+6x4162x+4 For Exercises 9-22, use long division to divide. (See Example 1-3) 60x2+8x41082x6 For Exercises 9-22, use long division to divide. (See Example 1-3) x5+4x4+18x220x10x2+5 For Exercises 9-22, use long division to divide. (See Example 1-3) x52x4+x38x18x23 For Exercises 9-22, use long division to divide. (See Example 1-3) 6x4+3x37x2+6x52x2+x3 18PE 19PE For Exercises 9-22, use long division to divide. (See Example 1-3) x3+64x+4 For Exercises 9-22, use long division to divide. (See Example 1-3) 5x32x2+32x1 For Exercises 9-22, use long division to divide. (See Example 1-3) 2x3+x2+13x+1 For Exercises 23-26, consider the division of two polynomials: fxxc. The result of the synthetic division process is shown here. Write the polynomials representing the a. Dividend. b. Divisor. c. Quotient. d. Remainder. 325542963630212101 24PE For Exercises 23-26, consider the division of two polynomials: fxxc. The result of the synthetic division process is shown here. Write the polynomials representing the a. Dividend. b. Divisor. c. Quotient. d. Remainder. 41225442441610 26PE For Exercises 27-38, use synthetic division to divide the polynomials. (See Example 4-5) 4x2+15x+1x+6 28PE For Exercises 27-38, use synthetic division to divide the polynomials. (See Example 4-5) 5x217x12x4 For Exercises 27-38, use synthetic division to divide the polynomials. (See Example 4-5) 2x2+x21x3 For Exercises 27-38, use synthetic division to divide the polynomials. (See Example 4-5) 48x3x25x4x+2 For Exercises 27-38, use synthetic division to divide the polynomials. (See Example 4-5) 5+2x+5x32x4x+1 For Exercises 27-38, use synthetic division to divide the polynomials. (See Example 4-5) 4x525x458x3+232x2+198x63x3 34PE For Exercises 27-38, use synthetic division to divide the polynomials. (See Example 4-5) x5+32x+2 For Exercises 27-38, use synthetic division to divide the polynomials. (See Example 4-5) x481x+3 For Exercises 27-38, use synthetic division to divide the polynomials. (See Example 4-5) 2x47x356x2+37x+84x32 For Exercises 27-38, use synthetic division to divide the polynomials. (See Example 4-5) 5x418x3+63x2+128x60x25 The value f6=39 for a polynomial fx . What can be concluded about the remainder or quotient of fxx+6?Given a polynomial fx, the quotient fxx2 has a reminder of 12. What is the value of f2?Given fx=2x45x3+x27, a. Evaluate f4 . b. Determine the remainder when fx is divided by x4.Given gx=3x5+2x4+6x2x+4, a. Evaluate g2 b. Determine the remainder when gx is divided by x2.For Exercises 43-46, use the remainder theorem to evaluate the polynomial for the given value of x. (See Example 6) fx=2x4+x349x2+79x+15a.f1b.f3c.f4d.f52 For Exercises 43-46, use the remainder theorem to evaluate the polynomial for the given value of x. (See Example 6) gx=3x422x3+51x242x+8a.g1b.g2c.g1d.g43 For Exercises 43-46, use the remainder theorem to evaluate the polynomial for the given value of x. (See Example 6) hx=5x34x215x+12a.h1b.h45c.h3d.h1 For Exercises 43-46, use the remainder theorem to evaluate the polynomial for the given value of x. (See Example 6) kx=2x3x214x+7a.k2b.k12c.k7d.k2 For Exercises 47-54, use the reminder theorem to determine if the given number c is a zero of the polynomial. (See Example 7) fx=x4+3x37x2+13x10a.c=2b.c=5 For Exercises 47-54, use the reminder theorem to determine if the given number c is a zero of the polynomial. (See Example 7) gx=2x4+13x310x219x+14a.c=2b.c=7 For Exercises 47-54, use the reminder theorem to determine if the given number c is a zero of the polynomial. (See Example 7) px=2x3+3x222x33a.c=2b.c=11 For Exercises 47-54, use the reminder theorem to determine if the given number c is a zero of the polynomial. (See Example 7) qx=3x3+x230x10a.c=3b.c=10 For Exercises 47-54, use the reminder theorem to determine if the given number c is a zero of the polynomial. (See Example 7) mx=x32x2+25x50a.c=5ib.c=5i 52PE 53PE For Exercises 47-54, use the reminder theorem to determine if the given number c is a zero of the polynomial. (See Example 7) fx=2x35x2+54x26a.c=1+5ib.c=15i For Exercises 55-60, use the factor theorem to determine if the given binomial is a factor of fx . (See Example 8) fx=x4+11x3+41x2+61x+30a.x+5b.x2 For Exercises 55-60, use the factor theorem to determine if the given binomial is a factor of fx . (See Example 8) gx=x410x3+35x250x+24a.x4b.x1 For Exercises 55-60, use the factor theorem to determine if the given binomial is a factor of fx . (See Example 8) fx=x3+64a.x4b.x+4 For Exercises 55-60, use the factor theorem to determine if the given binomial is a factor of fx . (See Example 8) fx=x481a.x3b.x+3 For Exercises 55-60, use the factor theorem to determine if the given binomial is a factor of fx . (See Example 8) fx=2x3+x216x8a.x1b.x22 60PE Given gx=x414x2+45, a. Evaluate g5. b. Evaluate g5. c. Solve gx=0 .Given hx=x415x2+44, a. Evaluate h11. b. Evaluate h11. c. Solve hx=0 .63PE 64PE a. Factor fx=2x3+x237x36, given that 1 is a zero. (See Example 9) b. Solve. 2x3+x237x36=0 a. Factor fx=3x3+16x25x50, given that 2 is a zero. b. Solve. 3x3+16x25x50=0 a. Factor fx=20x3+39x23x2, given that 14 is a zero. b. Solve. 20x3+39x23x2=0 a. Factor fx=8x318x211x+15, given that 34 is a zero. b. Solve. 8x318x211x+15=0 a. Factor fx=9x333x2+19x3, given that 3 is a zero. b. Solve. 9x333x2+19x3=0 70PE For Exercises 71-82, write a polynomial fx that meets the given conditions. Answers may vary. (See Example 10) Degree 3 polynomial with zeros 2,3,and4.72PE 73PE For Exercises 71-82, write a polynomial fx that meets the given conditions. Answers may vary. (See Example 10) Degree 5 polynomial with zeros 2,52 (each with multiplicity 1), and 0 (with multiplicity 3).75PE For Exercises 71-82, write a polynomial fx that meets the given conditions. Answers may vary. (See Example 10) Degree 2 polynomial with zeros 52and52.For Exercises 71-82, write a polynomial fx that meets the given conditions. Answers may vary. (See Example 10) Degree 3 polynomial with zeros 2,3i,and3i.For Exercises 71-82, write a polynomial fx that meets the given conditions. Answers may vary. (See Example 10) Degree 3 polynomial with zeros 4,2i,and2i.79PE For Exercises 71-82, write a polynomial fx that meets the given conditions. Answers may vary. (See Example 10) Degree 3 polynomial with integer coefficient and zeros of 25,32,and6.81PE For Exercises 71-82, write a polynomial fx that meets the given conditions. Answers may vary. (See Example 10) Degree 2 polynomial with zeros of 5+6iand56i.83PE 84PE a.Isx1afactorofx1001?b.Isx+1afactorofx1001?c.Isx1afactorofx991?d.Isx+1afactorofx991?e.Ifnisapositiveeveninteger,isx1afactorofxn1?f.Ifnisapositiveoddinteger,isx+1afactorofxn1?If a fifth-degree polynomial is divided by a second-degree polynomial, the quotient is a -degree polynomial.Determine if the statement is true or false: Zero is a zero of the polynomial 3x57x42x314.Determine if the statement is true or false: Zero is a zero of the polynomial 2x4+5x3+6x.Find m so that x+4 is a factor of 4x3+13x25x+m.Find m so that x+5 is a factor of 3x410x3+20x222x+m.Find m so that x+2 is a factor of 4x3+5x2+mx+2.92PE 93PE For what value of r is the statement an identify? x25x8x2=x3+rx2 provided that x2 A metal block is formed from a rectangular solid with a rectangular piece cut out. a. Write a polynomial Vx that represent the volume of the block. All distances in the figure are in centimeters. b. Use synthetic division to evaluate the volume if x is 6 cm.A wedge is cut from a rectangular solid. a. Write a polynomial Vx that represents the volume of the remaining part of the solid. All distances in the figure are in feet. b. Use synthetic division to evaluate the volume if x is 10 ft.Under what circumstances can synthetic division be used to divide polynomials?How can the division algorithm be used to check the result of polynomial division?Given a polynomial fx and a constant c, state two methods by which the value fc can be computed.Write an informal explanation of the factor theorem.a. Factor fx=x35x2+x5 into factors of the form xc, given that 5 is a zero. b. Solve. x35x2+x5=0 a. Factor fx=x33x2+100x300 into factors of the form xc, given that 3 is a zero. b. Solve. x33x2+100x300=0 a. Factor fx=x4+2x32x26x3 into factors of the form xc, given that 1 is a zero. b. Solve. x4+2x32x26x3=0 a. Factor fx=x4+4x3x220x20 into factors of the form xc, given that 2 is a zero. b. Solve. x4+4x3x220x20=0 For Exercises 105-106, a. Use the graph to determine a solution to the given equation. b. Verify your answer from part (a) using the remainder theorem. c. Find the remaining solutions to the equation. 5x3+7x258x24=0 For Exercises 105-106, a. Use the graph to determine a solution to the given equation. b. Verify your answer from part (a) using the remainder theorem. c. Find the remaining solutions to the equation. 2x3x241x+70=0 List all possible rational Zeros. fx=4x4+5x37x2+8 Find the zeros. fx=x3x24x2 3SP 4SP Given fx=x42x3+28x24x+52, and that 1+5i is a zero of fx, a. Find the zeros b. Factor fx as a product of linear factors. c. Solve the equation. x42x3+28x24x+52=0 a. Find a third-degree polynomial fx with integer coefficient and with zeros of 2+iand43. b. Find a polynomial gx of lowest degree with zeros of 3 (multiplicity 2) and 5 (multiplicity 2), and satisfying the condition that g0=450.Determine the number of possible positive and negative real zeros. fx=4x5+6x3+2x2+6 Determine the number of possible positive and negative real zeros. gx=8x85x7+3x5x23x+1 9SP Find the zeros and their multiplicities. fx=x5+6x32x227x18 The of a polynomial fx are the solutions (or roots) of the equation fx=0.If fx is a polynomial of degree n1 with complex coefficients, then fx has exactly complex zeros, provided that each zero is counted by its multiplicity.The conjugate zeros theorem states that if fx is a polynomial with real coefficient, and if a+bi is a zeros of fx , then is also a zero of fx .A real number b is called an bound of the real zeros of a polynomial fx if all real zeros of fx are less than or equal to b.A real number a is called a lower bound of the real zeros of a polynomial fx if all real zeros of fx are or equal to a.Explain why the number 7 cannot be a rational zero of the polynomial fx=2x3+5x2x+6.For Exercises 7-12, list the possible rational zeros. (See Example 1) fx=x52x3+7x2+4 8PE For Exercises 7-12, list the possible rational zeros. (See Example 1) hx=4x4+9x3+2x6 For Exercises 7-12, list the possible rational zeros. (See Example 1) kx=25x7+22x43x2+10 For Exercises 7-12, list the possible rational zeros. (See Example 1) mx=12x6+4x33x2+8 For Exercises 7-12, list the possible rational zeros. (See Example 1) nx=16x47x3+2x+6 13PE Which of the following is not a possible zero of fx=4x52x3+10? 3,5,52,32 15PE For Exercises 15-16, find all the rational zeros. qx=x4+x37x25x+10 17PE 18PE For Exercises 17-28, find all the zeros. (See Example 2-4) fx=x37x2+6x+20 For Exercises 17-28, find all the zeros. (See Example 2-4) gx=x37x2+14x6 21PE 22PE 23PE For Exercises 17-28, find all the zeros. (See Example 2-4) nx=2x49x35x257x45 For Exercises 17-28, find all the zeros. (See Example 2-4) qx=x34x22x+20 26PE For Exercises 17-28, find all the zeros. (See Example 2-4) tx=x4x290 For Exercises 17-28, find all the zeros. (See Example 2-4) vx=x412x213 Given a polynomial fx of degree n1, the fundamental theorem of algebra guarantees at least complex zero.30PE If fx is a polynomial with real coefficients and zeros of 5 (multiplicity 2), 1 (multiplicity 1), 2i,and3+4i, what is the minimum degree of fx ?If gx is a polynomial with real coefficients and zeros of 4 (multiplicity 3), 6 (multiplicity 2), 1+i,and27i, what is the minimum degree of gx ?For Exercises 33-38, a polynomial fx and one or more of its zeros is given. a. Find all the zeros. b. Factor fx as a product of linear factors. c. Solve the equation fx=0 (See Example 5) fx=x44x3+22x2+28x203;25iisazero For Exercises 33-38, a polynomial fx and one or more of its zeros is given. a. Find all the zeros. b. Factor fx as a product of linear factors. c. Solve the equation fx=0 (See Example 5) fx=x46x3+5x2+30x50;3iisazero For Exercises 33-38, a polynomial fx and one or more of its zeros is given. a. Find all the zeros. b. Factor fx as a product of linear factors. c. Solve the equation fx=0 (See Example 5) fx=3x328x2+83x68;4+iisazero 36PE 37PE For Exercises 33-38, a polynomial fx and one or more of its zeros is given. a. Find all the zeros. b. Factor fx as a product of linear factors. c. Solve the equation fx=0 (See Example 5) fx=2x55x44x322x2+50x+75;12iand52arezeros For Exercises 39-48, write a polynomial fx that satisfies the given conditions. (See Example 6) Degree 3 polynomial with integer coefficients with zeros 6iand45 For Exercises 39-48, write a polynomial fx that satisfies the given conditions. (See Example 6) Degree 3 polynomial with integer coefficients with zeros 4iand32 For Exercises 39-48, write a polynomial fx that satisfies the given conditions. (See Example 6) Polynomial of lowest degree with zeros of 4 (multiplicity 1), 2 (multiplicity 3) and with f0=160 For Exercises 39-48, write a polynomial fx that satisfies the given conditions. (See Example 6) Polynomial of lowest degree with zeros of 5 (multiplicity 2) and 3 (multiplicity 2) and with f0=450 For Exercises 39-48, write a polynomial fx that satisfies the given conditions. (See Example 6) Polynomial of lowest degree with zeros of 43 (multiplicity 2) and 12 (multiplicity 1) and with f0=16 For Exercises 39-48, write a polynomial fx that satisfies the given conditions. (See Example 6) Polynomial of lowest degree with zeros of 56 (multiplicity 2) and 13 (multiplicity 1) and with f0=25 45PE For Exercises 39-48, write a polynomial fx that satisfies the given conditions. (See Example 6) Polynomial of lowest degree with real coefficients and with zeros 510iand0 (multiplicity 3)For Exercises 39-48, write a polynomial fx that satisfies the given conditions. (See Example 6) Polynomial of lowest degree with real coefficients and with zeros 5iand6i.For Exercises 39-48, write a polynomial fx that satisfies the given conditions. (See Example 6) Polynomial of lowest degree with real coefficients and with zeros 3iand5+2i.For Exercises 49-56, determine the number of possible positive and negative real zeros for the given function. (See Example 7-8) fx=x62x4+4x32x25x6

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