MTH 65-Steiner Exam #1 Review: , , 8.6. Non-Calculator sections: (Solving Systems), Chapter 5 (Operations with Polynomials)

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1 Non-Calculator sections: (Solving Systems), Chapter 5 (Operations with Polynomials) The following problems are examples of the types of problems you might see on the non-calculator section of the test. 1. Solve each system by graphing. Describe why these problems are good graphing problems. Indicate whether there is one solution, no solution, or infinite solutions using set notation. 1 a) y = x + 1 b) y = x 4 y = x 5 x =. Solve each system by substitution. Describe why these problems are good substitution problems. Indicate whether there is one solution, no solution, or infinite solutions using set notation. a) 4x y = 10 b) x y = 6 y = x + 3 x = 13 y 3. Solve each system by addition. Describe why these problems are good addition problems Indicate whether there is one solution, no solution, or infinite solutions using set notation. a) 3x 4 y = 1 b) 5x y = 8 6 x + 8 y = 3x 5 y = 1 4. Solve each system by a method of your choice. Write the method chosen and why. Indicate whether there is one solution, no solution, or infinite solutions using set notation. a) 3 x + 4 y = 8 b) y 7 = 0 c) x 3y 4 = x + 3 y = 5 7x 3 y = 0 x + y 4 = 7 5

2 5. Perform the indicated operations. You may have to combine like terms, distribute, FOIL, and/or use a shortcut formula. Write the method you use for each problem and why. a. ( 9x 3 7x + 5 ) + ( 4x 3 x + 7x 10) b. 3x ( 7x + 3x 6 ) c. ( 13x3y 5x y 9x ) ( 11x3y 6x y 3x + 4) d. ( x + 3)(x 5 x + ) e. ( x + 6 )(x + ) f. ( 5x + 4)(5x 4) g. (x + 3) h. (y 1) i. ( x 3 )(x + ) j. ( 4x 3)(3x 1) k. (3x + 4) l. (x 6 )(x + 7) m. x(x 7) n. 3x(x 5 )(x + 3 ) 6. Simplify using the rules of exponents. a. 5 x 10x 3 b. (10y) c. (x 5) 7 d. ( 10x y)(5xy) e. 6x y(5x 9y) f. (x 4 y) g. ( 3x 3 ) h. x x 4

3 Calculator: The following problems are examples of the types of problems you might see on the calculator section of the test. (4.4 Systems Word Problems, 8.6 Functions, Ch. 5 Word Problems) 1. On a special day, tickets for a minor league baseball game cost $5 for adults and $1 for students. The attendance that day was 181 and $345 was collected. Find the number of each type of ticket sold. Step 1 (define variables): Step (write equations): Step 3 (solve/interpret):. You are choosing between two plans at a discount warehouse. Plan A offers an annual membership fee of $300 and you pay 70% of the manufacturer s recommended list price. Plan B offers an annual membership fee of $40 and you pay 90% of the manufacturer s recommended list price. How many dollars of merchandise would you have to purchase in a year to pay the same amount under both plans? What will be the cost of each plan? Step 1 (define variables): Step (write equations): Step 3 (solve/interpret):

4 3. Functions f (x), g (x), and h (x) are pictured below. Use them to answer the questions. 1 h (x) = x 3 x f (x) g (x) (above) Write your answers as points (x,y) AND in function notation. a. Evaluate f( 1) b. Evaluate g() c. Evaluate h (8) d. Solve f (x) = 1 e. Solve g (x) = f. Solve h (x) = 4 g. State the domain and range for f(x) h. State the domain and range for g (x) 4. The postal Service charges $.3 for the first ounce to mail a first class letter. It charges $.3 for each additional ounce. An equation that can be used to model this situation is: c (w) = w. a. Define the variables w and c(w) in terms for the situation. b. Evaluate c (5) and interpret the solution in terms of the situation. c. Solve c (w) = 8 and interpret the solution in terms of the situation. Round to the nearest tenth. d. Another equation than can be used to model this situation is c (w) = (w 1 ). Define the variable w for this equation.

5 5. The square painting in the figure measures x inches on each side. The painting is uniformly surrounded by a frame that measures 1 inch wide. a. Write an expression that represents the length of the square that includes the painting and the frame. b. Write an expression that represents the width of the square that includes the painting and the frame. c. Use the results from a and b to write a polynomial that represents the area of the square that includes the painting and the frame. d. Use the result from c to calculate the wall area needed to hang paintings with x = 0 in, x = 5 in and x = 8 in. e. Write an expression that describes the area of the frame. f. Use the result from e to calculate the amount of material needed to make a frame for paintings with x = 0 in, x = 5 in and x = 8 in. 6. The polynomial s 3 70s s 10, 800 models the profit a company makes on selling an item at a price s. A second item sold at the same price brings in a profit of s 3 30s + 450s a. Write a polynomial that expresses the total profit from the sale of both items. b. Use the polynomial from part a to evaluate the profit from both items if the sales price was $50.

6 Solutions Non Calculator Section 1. a. {(4, -1)} b. {(-, -8)}. a. No solution b. {(5, 4)} 3. a. Infinite Solutions { (x, y) 3x 4 y = 1 } b. {(, 1)} 4. a. {(-4, 1)} b. {(3, 7)} c. {(10, 8)} 5. a. 13x 3 8x + 7x 5 b. -1x 4 + 9x 3-18x c. x 3 y + x y - 6x - 4 d. x 3 - x - 13x + 6 e. x + 8x + 1 f. 5x - 16 g. x + 6x + 9 h. y - y + 1 i. x + x 6 j. 1x 13x + 3 k. 9x + 4x + 16 l. 4x 10x 84 m. x 7x n. 3x 3 6x 45x 6. a. 50x 4 b. 100y c. x 35 d. 50x 3 y e. 30x 4 y + 54x y f. x 8 y 4 g. 9x 6 h. 3x 4 + 7

7 Calculator 1. x represents the number of adult tickets and y represents the number of student tickets x + y = 181 the first equation is the total sales 5 x + y = 345 the second equation is the money earned The solution is {(536,745)} and represents 536 adult tickets and 745 student tickets sold.. x represents the yearly $ amount purchased and y represents the annual cost of a plan y = x the first equation is Plan A y = x the second equation is Plan B The solution is {(1300, 110)} and represents $1300 of merchandise for a cost of $ a. (-1, 5) AND f( 1 ) = 5 b. (, 1) AND g () = 1 c. (8, 1) AND h (8) = 1 d. (3, -1) & (, -1) AND f (3) = 1 & f() = 1 e. f. (14, 4) AND h (14) = 4 g. D: {3, -1,, 0}; R: {-1, 5, -3} h. D: (-, ); R: [0, ) 4. a. w represents the additional weight (over 1oz) of a letter in ounces. c (w) represents the cost to mail a letter in dollars. b. c (5) = 6.07 ; the cost to mail a 5+1 = 6 oz letter is $6.07 c. c (33.4) = 8 ; it costs $8 to mail a = 34.4 oz letter d. Now, w represents the weight of the letter in oz. 5. a. x + in b. x + in c. x + 4 x + 4 in d. 4 in, 49 in, 100 in e. 4 x + 4 in f. 4 in, 4in, 36 in 6. a. s 3 100s s b. A sales price of $50 is a $81,700 profit from both items.