Simplifying Algebraic Expressions

Math 110 Final Exam – Form A Name: _____________________________ Directions: 1. Show all work to receive full credit and clearly box your final answers. 2. Please turn in all scratch paper at the end of the exam. 3. Please make sure your cell phone is off and out of sight, unless you are using it to access the exam. 4. You may use a graphing calculator during the exam. 5. Please make sure your cameras are on and you are visible the whole time to get credit for the exam (see detailed instructions on Canvas). It has truly been a pleasure having you in my class. Good luck and enjoy your summer break! I. Simplifying Algebraic Expressions. (4 pts each) Perform the indicated operation and simplify. Assume variables represent nonnegative numbers. Write any complex solutions in the form a+bi. 1. (5 βˆ’ 4𝑖)(3 + 2𝑖) 2. 3 π‘₯βˆ’2 βˆ’ 2 π‘₯+1 π‘₯ +4π‘₯+3 3. Rationalize the denominator: 4. (2π‘₯ βˆ’ 5𝑦)! ( √*+, II. Solving Equations (4 pts each) Solve the following equations. Round your answer to two decimal places (nearest hundredth). 5. π‘₯ ! + 6π‘₯ βˆ’ 2 = 0 8. π‘₯ βˆ’ 4 = √2π‘₯ βˆ’ 8 6. log(π‘₯) + log(π‘₯ βˆ’ 15) = 2 7. 2π‘₯ + 5𝑦 = 10 3π‘₯ βˆ’ 4𝑦 = 15 (use elimination or substitution) 9. 3″#! = 4 10. . ” /0. βˆ’ * ./0 = ( . 11. π‘₯ $ βˆ’ 13π‘₯ ! + 36 = 0 (use u substitution) 12. |2π‘₯ βˆ’ 3| β‰₯ 7 (write solution in interval notation) III. Application Problems. (5 pts each) Please translate each of the application problems into mathematical equations and solve using any method of your choice. 13. Find the length of the time for $10,000 to double when it is invested at 6% annual interest rate, compounded continuously. Use the formula A = Pe rt . 14. If you choose to use matrices using a graphing calculator, please write the augmented matrix that you will enter on your calculator to solve the problem. An electrician, a carpenter, and a plumber are hired to work on a house. The electrician earns $30 per hour, the carpenter $28.50 per hour, and the plumber $34 per hour. The first day on the job, they worked a total of 21.5 hours and earned a total of $673.00. If the plumber worked 2 more hours than the carpenter did, how many hours did each work? IV. Graphing (4 pts each). Graph the following and find any requested information. 15. y = ex βˆ’ 3 Domain (in interval notation): __________ Range (in interval notation): __________ 16. 2 y = ( x βˆ’ 2) + 3 Vertex: _______ Line of symmetry: ___________ 17. (π‘₯ + 1)! + (𝑦 βˆ’ 3)! = 16 Find the center of the circle: _______ Find the radius of the circle: _______ 18. ( x βˆ’ 2) 4 2 ( y βˆ’ 3) βˆ’ 9 2 =1 Center: _______ Show the asymptotes on the graph. 19. 25x 2 + y 2 = 25 Equation in standard form: _______________________ Center of this ellipse: ________ V. Other problems (3 pts each) 20. Perform the given row operation indicated write out the new matrix. ” 2 βˆ’1 1 βˆ’1 % $ ‘ $ 1 βˆ’3 4 βˆ’2 ‘ $ 4 βˆ’2 1 βˆ’6 ‘ # & βˆ’2R1 + R3 β†’ R3 Resulting Matrix: 21. Divide using long division or synthetic division. 2π‘₯ ! βˆ’ 3π‘₯ + 5 Γ· π‘₯ βˆ’ 4 22. Write in expanded form and find the sum: 4 Γ₯n 2 +2 n =1 23. Find the 10th term a10 of the sequence 2, 6, 18, 54, … 24. Find the partial sum S7 of the sequence 3, 8, 13, 18, … 25. Find the inverse of f ( x ) = 3x +1 5 26. Given the following function g ( x), find the following (4 points). g ( x) π‘₯ value(s) for which 𝑔(π‘₯) = 5: ________ (𝑔 ∘ 𝑔)(2): ______ BONUS QUESTIONS (3 points each): 1. Given the equation 9𝑦 ! βˆ’ 36𝑦 βˆ’ π‘₯ ! βˆ’ 6π‘₯ + 18 = 0 of a hyperbola, write in standard form. 2. Solve the system of nonlinear equations by substitution or elimination. x 2 + y 2 = 10 2x 2 βˆ’ y 2 = 17
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