Name: Class: Date: ID: A Algebra Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Determine which binomial is not a factor of 4x 4 1x 3 46x + 19x + 180. a. x + 4 c. x 5 b. x + 3 d. 4x + 3. Determine which binomial is a factor of x 3 + 6x 5x 6. a. x 6 b. x + c. x 5 d. x Short Answer 3. Use a graphing calculator to find a polynomial function to model the data. x 1 3 4 5 6 7 8 9 10 f(x) 1 4 5 13 9 16 19 16 4 43 4. The table shows the number of hybrid cottonwood trees planted in tree farms in Oregon since 1995. Find a cubic function to model the data and use it to estimate the number of cottonwoods planted in 006. Years since 1995 1 3 5 7 9 Trees planted (in thousands) 1.3 18.3 70.5 177.1 357.3 5. The table shows the number of llamas born on llama ranches worldwide since 1988. Find a cubic function to model the data and use it to estimate the number of births in 1999. Years since 1988 1 3 5 7 9 Llamas born (in thousands) 1.6 0 79. 03. 416 6. Write the expression (x )(x + 4) as a polynomial in standard form. 7. Miguel is designing shipping boxes that are rectangular prisms. One shape of box with height h in feet, has a volume defined by the function V(h) = h(h 5)(h 9). Graph the function. What is the maximum volume for the domain 0 < h < 9? Round to the nearest cubic foot. 8. Use a graphing calculator to find the relative minimum, relative maximum, and zeros of y = 3x 3 + 15x 1x 60. If necessary, round to the nearest hundredth. 9. Find the zeros of y = x(x + )(x + 3). Then graph the equation. 10. Write a polynomial function in standard form with zeros at 3, 5, and 1. 1
Name: ID: A 11. Divide 4x 3 x + x + 4 by x 3. Divide using synthetic division. 1. (x 4 x 3 x + 68x 3) (x + 4) 13. (x 3 + 4 11x + 3x ) (6 + x) 14. Use synthetic division to find P( 1) for P(x) = x 4 + x 3 + 8x + 10x 5. Solve the equation by graphing. 15. x + 3x + 3 = 0 16. x 3 x 0x = 0 17. 6x = 9 + x Factor the expression. 18. x 3 + 15 19. c 3 51 0. x 4 5x + 576 1. Solve x 3 + 343 = 0. Find all complex roots.. Solve x 4 45x = 34. 3. Use the Rational Root Theorem to list all possible rational roots of the polynomial equation x 3 + x + x + 9 = 0. Do not find the actual roots. 4. Find the rational roots of x 4 + 8x 3 + 7x 40x 60 = 0. Find the roots of the polynomial equation. 5. x 3 x + 10x + 136 = 0 6. x 3 + x 19x + 0 = 0 7. x 4 5x 3 + 11x 5x + 30 = 0 8. A polynomial equation with rational coefficients has the roots + 7, 7. Find two additional roots.
Name: ID: A 9. Find a third-degree polynomial equation with rational coefficients that has roots 6 and 3 + i. 30. Find a quadratic equation with roots 1 + 4i and 1 4i. 31. Find all zeros of x 4 5x 3 + 53x 15x + 75 = 0. 3. The table shows the population of Rockerville over a twenty-five year period. Let 0 represent 1975. Population of Rockerville Year Population 1975 336 1980 350 1985 359 1990 366 1995 373 000 395 a. Find a quadratic model for the data. b. Find a cubic model for the data. c. Graph each model. Compare the quadratic model and cubic model to determine which is a better fit. Essay 33. Find the rational roots of 4x 3 3x 1 = 0. Explain the process you use and show your work. Other 34. What are multiple zeros? Explain how you can tell if a function has multiple zeros. 35. Use division to prove that x = 3 is a real zero of y = x 3 + 9x 38x + 60. 36. A polynomial equation with rational coefficients has the roots 7 and 3. Explain how to find two additional roots and name them. 3
ID: A Algebra Review Answer Section MULTIPLE CHOICE 1. A. D SHORT ANSWER 3. f(x) = 0.08x 4 1.73x 3 + 1.67x 34.68x + 35.58 4. T(x) = 0.4x 3 + 0.8x + 0.1x; 630.3 thousand trees 5. L(x) = 0.5x 3 + 0.6x + 0.3x + 0.; 741,600 llamas 6. x + x 8 7. 4 ft 3 8. relative minimum: (0.36, 6.4), relative maximum: ( 3.69, 37.79), zeros: x = 5,, 9. 0,, 3 10. f(x) = x 3 3x 13x + 15 11. 4x + 11x + 34, R 106 1. x 3 5x + 19x 8 13. x 3x + 7, R 38 14. 7 15. no solution 16. 0, 3.4,.9 17. 3 18. (x + 5)(x 5x + 5) 19. (c 8)(c + 8c + 64) 0. (x 4)(x + 4)(x 6)(x + 6) 1
ID: A 1. 7, 7 ± 7i 3. 6, 6, 3, 3 3. 9, 3, 1, 1, 3, 9 4. 6, 5. 3 ± 5i, 4 3 + i 6., 3 i, 4 7., 3, ± i 5 8. 7, 7 + 9. x 3 6x + 60 = 0 30. x + x + 17 = 0 31. 1, 3, ± 5i 3. a. y = 0.03x + 1.549x + 338.571 b. y = 0.0079x 3 0.716x + 4.378x + 335.670 c. The cubic model is a better fit.
ID: A ESSAY 33. [4] Step 1: List the possible rational roots by using the Rational Root Theorem. The leading coefficient is 4 with factors of ±1, ±, and ±4. The constant term is 1 with factors factor of of 1 and 1. The only possible roots of the equation have the form 1 factor of 4. Those roots would be ±1, ± 1, and ± 1 4. Step : Test each possible rational root in the equation. The only roots that satisfy the equation are 1 and 1. [3] an error in computation or missing part of the explanation [] several errors in computation or in the explanation [1] one root given with no explanation OTHER 34. If a linear factor of a polynomial is repeated, then the zero is repeated and the function has multiple zeros. To determine whether a function has a multiple zero, factor the polynomial. If a factor is repeated in the factored expression, then it is a multiple zero. 35. x 3 + 9x 38x + 60 (x 3) = x + 6x 0 with no remainder, so x = 3 is a real zero of the function. 36. By the Irrational Root Theorem, if 7 is a root, then its conjugate 7 is also a root. If 3 is a root, then its conjugate 3 is also a root. Two additional roots are 7 and 3. 3