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1 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 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 f(x) The table shows the number of hybrid cottonwood trees planted in tree farms in Oregon since Find a cubic function to model the data and use it to estimate the number of cottonwoods planted in 006. Years since Trees planted (in thousands) The table shows the number of llamas born on llama ranches worldwide since Find a cubic function to model the data and use it to estimate the number of births in Years since Llamas born (in thousands) 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 x 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
2 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 x + 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 = x 3 x 0x = x = 9 + x Factor the expression. 18. x c x 4 5x Solve x = 0. Find all complex roots.. Solve x 4 45x = 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 = 0 6. x 3 + x 19x + 0 = 0 7. x 4 5x x 5x + 30 = 0 8. A polynomial equation with rational coefficients has the roots + 7, 7. Find two additional roots.
3 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 x 15x + 75 = The table shows the population of Rockerville over a twenty-five year period. Let 0 represent Population of Rockerville Year Population 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 A polynomial equation with rational coefficients has the roots 7 and 3. Explain how to find two additional roots and name them. 3
4 ID: A Algebra Review Answer Section MULTIPLE CHOICE 1. A. D SHORT ANSWER 3. f(x) = 0.08x x x 34.68x T(x) = 0.4x x + 0.1x; thousand trees 5. L(x) = 0.5x x + 0.3x + 0.; 741,600 llamas 6. x + x ft 3 8. relative minimum: (0.36, 6.4), relative maximum: ( 3.69, 37.79), zeros: x = 5,, 9. 0,, f(x) = x 3 3x 13x x + 11x + 34, R x 3 5x + 19x x 3x + 7, R no solution 16. 0, 3.4, (x + 5)(x 5x + 5) 19. (c 8)(c + 8c + 64) 0. (x 4)(x + 4)(x 6)(x + 6) 1
5 ID: A 1. 7, 7 ± 7i 3. 6, 6, 3, , 3, 1, 1, 3, , 5. 3 ± 5i, i 6., 3 i, 4 7., 3, ± i , x 3 6x + 60 = x + x + 17 = , 3, ± 5i 3. a. y = 0.03x x b. y = x x x c. The cubic model is a better fit.
6 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
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