Polynomial Unit Review Name _________________________________ 1. The diagram below shows a pool surrounded by a walkway. All units are meters. a) What is the expression for the area of the pool? b) What is the expression for the area of the pool plus the walkway c) If the walkway will be built from tiles that are 1 meter by 1 meter, what expression can we use for the number of tiles needed to build the walkway? d) If the pool is 20 meters by 10 meters, how many tiles will be needed for the walkway? 2. Dots – the first three dot figures in a pattern are shown below. a) How many dots would be in the fourth and fifth figures? b) What is the expression for the number of dots in the nth pattern? c) Would any figure in the pattern have exactly 204 dots? Explain your answer. 2. Put each of the following into standard form and indicate the degree of the polynomial. a) 2x3 + x2 – 5x – 1 + (3x2 – 2x + 5) b) 75pq + 2p – 4q – 2(13pq – 5p + 7q) c) 5x2y – 10xy2 + 9xy + (4xy3 – 2x2y + 11) d) (3x – 5)(x2 + 7) 3. Acme Toys has calculated that its earnings, in thousands of dollars, from selling its Justin Bieber action figures is given by the expression -‐x2 + 7x + 30, where x is the number of units, in the thousands, that Acme produces. The company has also calculated that the cost of making the action figures is given by the expression x + 30 The profit the company will earn is equal to the earnings minus the cost. What is the expression for the Acme’s profit from the Justin Bieber action figure? a) What two values of x will set the total profits to zero? b) What value of x will give the highest possible profit? c) Find the total earnings, total cost, and total profit if Acme produces the number of units that produces the highest profit (plug your answer from the previous question into the three expressions). 4. Use multiplication boxes to find the following products, in fully simplified form. a) 432 x 27 400 30 2 20 7 b) (x2 + 3x – 5)(x – 2) x2 +3x -‐5 x -‐2 c) (2x2 + 5xy + y2)(x + 3y) 2x2 5xy y2 x 3y d) (2x2 – 5)(7x + 4) [create your own table] e) (x + 3)3 [create your own tables – this will require multiplication in two steps] 5. Jean Claude Killy plans to construct a vegetable garden outside his L-‐shaped house. He has purchased 40 feet of wire fencing to build 2 sides of a rectangular enclosure (the house will serve as the other 2 sides). a) If the width of the garden is w, what expression (using w) represents the length of the garden? b) What expression represents the total area of the garden? c) What dimensions will create the garden with the greatest possible area? 6. Find all solutions for the following. Put your answer in the form of a solution set. a) 2x(x – 3)(x + 1) = 0 b) (3x – 2)(2x – 3) = 0 c) 4x2 – 20x = 0 e) (x + 2)2 – 9 = 0 d) 3x3 – 27x = 0 7. Factor each of the following, if possible. If the expression cannot be factored, write “Prime” next to it. Example 1: x3 – 6x2 Can be factored: x2(x – 6) Example 2: x3 – 4x2 – 12x Can be factored: x(x2 – 4x – 12) x2 – 4x – 12 can be factored to (x – 6)(x + 2) Final factorization: x(x – 6)(x + 2) Example 3: x2 – 5x – 2 Cannot be factored: no common factors and x2 – 5x – 2 cannot be expressed as product of factors a) 4x2 – 24x b) x2 – 3x – 40 c) x3 + 6x2 + 9x d) x2 – y2 e) x4 – y4 f) x2 – 8x + 16 g) 3x4 + 9x3 + 15x2 h) x2 – 7x – 2 8. Multiply the following using the repeated distributive property method: Example: (x2 – 2x + 1)(3x + 5) = x2(3x + 5) – 2x(3x + 5) + 1(3x + 5)= 3x3 + 5x2 – 6x2 – 10x + 3x + 5 = 3x3 – x2 – 7x + 5 a) (2x – 6)(4x + 3) b) (x2 + 2x + 1)(x + 1) c) (x – 1)(x2 + x + 1) d) (x – 4)(x + 5) 9. Find the solution set for each of the following, using factoring if appropriate. a) x2 – 5x – 6 = 0 b) x3 – 9x = 0 c) x2 – 2x = 24 (set right side equal to d) 2x2 – 6x – 20 = 0 (factor out common zero first) factor first) e) x2 = 64 f) (x – 2)2 = 16 g) (x – 4)2 = 10 h) x2 + 6x + 9 = 25 10. Find the value of a in each of the following if each expression is a perfect square trinomial. A perfect square trinomial has the form (x + k)2 = x2 + 2kx + k2, where k is an integer, positive or negative. a) x2 + 6x + a b) x2 – ax + 25 c) x2 – 12x + a2 d) x2 – 5ax + 25 e) 4x2 – ax + 9 f) x2 + 2ax + 49 11. Provide three examples of expressions that are the difference of perfect squares. Then factor each expression. One of your expressions must have no coefficients equal to 1. Example: x2 – 49 = (x – 7)(x + 7) 12. The height off the ground of a bowling ball dropped from the top of a 640-‐foot tall building is given by the equation: h(t) = -‐16t2 + 640, where t is the number of seconds since the ball was dropped a) What is the height of the ball after 5 seconds? b) How long will it take the ball to hit the ground? 13. If the ball in the above situation is not dropped, but thrown upward with an initial velocity of 96 feet per second the equation giving its height becomes: h(t) = -‐16t2 + 96t + 640 a) When will the ball hit the ground (hint: set h(t) = 0 and then divide both sides by -‐16). Use factoring. b) Another form of the equation is h(t) = -‐16(t – 3)2 + 784. Use this form to find the maximum height the ball will reach and how long it will take to reach that height. c) How long after being thrown into the air will it take for the ball to reach a height of 640 feet again? (Set h(t) to 640 and solve). 14. Solve each of the following using the completing the square method: a) x2 + 10x + 40 = 51 b) x2 + 4x + 2 = 8 c) x2 + 8x = 16 d) x2 – 12x + 21 = 0