Name Algebra II Date Period Unit 4 Test REVIEW: Polnomial Functions 1. Given a polnomial of the form: = a n + b n 1 + c n 2 + + d 2 + e + f a. What are the maimum number of zeros for this polnomial? b. What are the minimum number of zeros for this polnomial if the highest eponent, n, is odd? c. What are the minimum number of zeros for this polnomial if the highest eponent, n, is even? 2. What is the remainder when P() = 4 3 3 2 + 5 is divided b + 2? a. 0 b. 17 c. -5 d. -51 3. Prove the identit ( ) 3 = 3 3 2 + 3 2 3 4. If the point (3, -5) lies on the graph of a power function with an odd eponent, which of the following points must also lie on its graph? a. (-3, 5) c. (-5, 3) b. (-3, -5) d. (3, 5) 5. A rectangular bo has a height of feet. The width is 2 less than its height and the length is 3 more than twice its height. If the volume of the bo is 27 cubic feet, use a graph to find the dimensions of the bo. Be sure to define our variables. f() 1
6. Solve the given equation for a: 3 2 a a a 6 12 72 7. Given the following rational epression: a. Simplif using long division. 3 9 10 +1 b. Epress the answer in quotient-remainder form. c. Is + 1 a factor of 3 9 10? d. How could ou determine if + 1 is a factor without using long division? 8. a. Create the equation of a cubic polnomial, in standard form, that has -intercepts given b the set {-5, 1, 6} and passes through the point (2, -14). b. Verif our answer b sketching the cubic s graph on the aes below. 2
2 2 2 2 2 2 2 9. A famous identit that can be used to general pthagorean triples is: 2 (a) Prove this identit.. (b) Generate the sides of a right triangle if = 3 and = 2. Show these sides satisf the Pthagorean Theorem. 10. Solve: 2 4 ( 3 1)( 9) 0 11. The graph below represents the long run behavior for which of the following functions? (Please tr this without graphing each of the following equations) a. 2 2 3 4 c. 4 9 5 3 1 b. 3 2 3 4 7 5 d. 5 3 2 5 8 7 2 6 12. The formula for the volume of a square pramid is V = 1 Bh where B is the area of the square base and h is the 3 height of the pramid. If the volume is represented b 1 3 (3 + 8 2 + 21 + 18) and the height of the pramid is + 2, what epression represents the length of a side of the base of the pramid? 13. Solve the following equation for : 5 3 4 3 + 3 2 2 + 6 = 0 3
14. The total profit (in thousands of dollars), P, that a compan makes for producing makeup kits (in thousands) can be modeled b the function: = 4 3 + 25. Construct the graph of this function in order to answer questions a-f. a. How man kits, to the nearest hundred, should the compan make to maimize their profit? b. What is the maimum profit to the nearest hundred dollars? c. For what number of makeup kits produced is the profit decreasing? (Round the boundaries of the interval to the nearest hundred) d. What is the average rate of change in the profit when the compan goes from making 1200 kits to 2400 kits? e. For what numbers of makeup kits produced does the compan make mone? f. Currentl the compan makes 2,000 kits and makes a profit of $18,000. Use our graph to determine a lesser number of kits the compan could produce and still make the same profit. Round to the nearest hundred kits. 15. Which of the following epressions is a factor of Q() = 3 2 2 15 14? a. 5 b. + 5 c. 2 d. + 2 16. Solve the following equation algebraicall: (4 2 + 3) 2 3(4 2 + 3) 4 = 0 4
17. Given the following cubic function: f() = 3 2 10 8 a. If f(4) = 0, use long division to completel factor f(). 25 b. What are the roots of f()? Verif our answer with a graph. -25 18. For each of the following polnomials, write a power function that best represents its end behavior. Then sketch the end behavior of each power function: a. Polnomial Power Function Sketch of Power Function 3 2 3 4 2 1 b. 2 2 5 2 c. 5 3 4 7 6 9 d. 2 4 8 3 1 5
19. Consider the quartic whose equation is 2 13 14 24 a. Sketch the graph of this function on the aes given below. Clearl label our window, and intercepts, and all relative etrema. If necessar, round to the tenths place. b. Use our graph from part a to find the solutions to the equation: 2 13 14 24 0 c. Considering our answer to part b, epress the equation 2 13 14 24 0 in completel factored form. 20. If a, b, and c are all positive real numbers, which graph could represent the sketch of the graph of p() = a( b)( 2 + 2c + c 2 )? 6