DMA 060 Worksheet 1 ~ Rules of Exponents I.
Rules of Exponents 1. Suppose you were given the product (x7) (x5). a. Expand x7 b. Expand x5 c. What is the expanded form of (x7) (x5)? d. What is the product written with exponents 2. Simplify the following using the Product Rule a. (x3)(x2) b. !! ∙ !! c. 12! ∙ 12! d. (!! ! ! )(!! ! ! ) e. ! + ! ! (! + !)! !!
3. Suppose you were given the quotient ! ! a. Expand x7 b. Expand x3 !!
c. What is the expanded form of ! ! d. What is the quotient written with exponents? 4. Simplify the following using the Quotient Rule. (Assume there is no division by zero) a.
b.
c.
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5. Use your calculator to fill in the blanks. Leave your answer as a whole number or a fraction. 33 = ____________________ 43 = ____________________ 32 = ____________________ 42 = ____________________ 31= _____________________ 41 = ____________________ 30 = _____________________ 40 = ____________________ 3-‐1 = ____________________ 4-‐1 = ____________________ 3-‐2 = ____________________ 4-‐2 = ____________________ 3-‐3 = ____________________ 4-‐3 = ____________________ What is the result when the exponent is 0? What happens when the exponent is negative? Is there a number where a = a-‐1? 6. Write each expression without using a negative exponent. a. x-‐2 b. (2y)-‐3 c.
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7. Suppose you were asked to simplify (!")! a. Expand (!")! b. How many factors of a are there? How many factors of b are there? c. How can you write the expanded form in exponential form different from the original problem? 8. Simplify each expression. a. (4p3)3 b. (a3b2)5 c. (2x2y3)3 d. (-‐3x2y4)2 9. Using the rules for exponents simplify each of the following. Do not leave parentheses or negative exponents in any of the answers. f. (c2d3)-‐3 a. ! ! ! ∙ !! ! b. (3! ! ! ! )! g. (-‐3v-‐2u3)-‐3 c.
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DMA 060 Worksheet 2 ~ Polynomial Operations Vocabulary. Fill in the blank (Words may be used more than once) Coefficient Trinomial Leading Polynomial constant largest Smallest binomial Monomial 0 1 2 A ________________________ is a term or sum of terms in which all variables have whole number exponents. The numerical _____________________ of the term – 25x2y3 is -‐25 The degree of a polynomial is the same as the degree of its term with the ____________ exponent. A __________________ is a polynomial with one term. A __________________ is a polynomial with two terms. A __________________ is a polynomial with three terms. For the polynomial 6x2 + 3x – 1, the __________________ term is 6x2, and the leading _____________________ is 6. The ________________________ term is -‐1. The degree of the polynomial is ___________________. We call this polynomial a __________________________. Adding and Subtracting Polynomials 1. 3x + 2x 2. 3x2 + 2x2 3. 5x3 – x3 4. 2x – 3y + 5x – y 5. 5x2 + (2y – 6x2) 6. (18y2 – 5) +(-‐3y2 – y + 7) 7. (8x2 – 5) +(-‐3y2 – y + 7) 8. (8x2 + 3x -‐4) – (-‐2x2 + 3x – 5) 9. (2x2 + 9) subtracted from (-‐6x2 + 9) 10. Find the perimeter of each figure a. (x2 +3x +1)yd (x2+3x + 1)yd (x2-‐4)yd b. (2x2-‐ 7) mm (2x – 5) mm (x + 6) mm (5x2 + 3x + 1) mm 11. Two warning flares are fired upward at the same time from different parts of a ship. The height in feet of the first flare is given by the polynomial function ! ! = −16! ! + 128! + 20 and the height in feet of the higher-‐traveling second flare is given by the polynomial function ! ! =
−16! ! + 150 ! + 40, after t seconds. a. Find a polynomial function h(t) that represents the difference in heights of the two flares b. In 4 seconds, the first flare reaches its peak, explodes, and lights up the sky. How much higher is the second flare at that point. III.
Multiplying Polynomials. 1. Multiply the monomial and the polynomial a. 3y ( x + 4y) b. 2ab2(2a + 3b – 2a2) c. -‐8x3(3x – 5y) 2. Multiply the binomials a. (a + 4)(a + 5) b. (2x – 2)(x – 4) c. (3x – 5)(2x + 1) d. (3y + 7)(2y – 5) e. (3x + 4)2 f. (y – 5)2 g. (4x + 5)(4x – 5) h. (2x – 1)(2x + 1) 3. Multiply the binomial and the trinomial a. (x – 5) (x2 + 2x – 3) b. (3x + 5) (2x2 – 3x + 1) c. (4x2 + 3x – 4) (3x + 2) 4. Find the area of the parallelogram or rectangle. a. (2x + 3) ft (8x + 1) ft b. (2x + 1) cm (3x – 4)cm IV.
Dividing Polynomials 1. Divide and simplify a.
b.
c.
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2. Applications a. What binomial when divided by 15n2, yields 2n2 – n? b. What trinomial when divided by 9a4b, yields 9a4-‐ 3a2b + 5a? c. The area covered by the rectangle below is (3x3 – 6x) square feet. Find an expression for the missing side. 3x ft DMA 060 Worksheet 3 Factoring Polynomials I.
Factor the GCF 1. 12x – 18y 2. 8x3 – 9x2 + 15x 3. 14xy – 16x2 y2 4. 5(4x – 3)3 + 2(4x – 3)2 II.
Factor by grouping 1. 2x + 2y + ax + ay 2. 6a3 – a2 + 18a – 3 3. 6x2 – 2x – 15x + 5 III.
Factoring Trinomials 1. x2 – 12x + 35 2. x2 + 5x + 6 3. X2 – 14X – 15 IV.
Factoring polynomials – Factor completely 1. 9x2 + 30x + 15 2. 4x2 -‐ 9 3. x2 + 36 4. 16x2 – 64 5. 5x3 – 30x2 – 35x 6. 6x2 – 7xy – 5y2 7. 2x – 10 + 3ax – 15a 8. 16y2 – 20xy + 25y2 9. X3 + 2X2-‐ 5X – 10 10. 2n2 -‐ 38n + 80 11. 2x2 + 11x + 15 12. 12x3y – 22x2y + 8xy 13. 16x2+ 24xy + 9y2 V.
Applications 1. The area of a rectangle is (4x2 + 20x – 11) in3. Factor the expression to find the dimensions of the rectangle. 2. The volume of the 8-‐inch wide box shown below is given by the expression (72x2+ 120x – 400)in2. If its dimensions can be determined by factoring the expression, find the height and the length of the container. 8 in ? ? 3. A student begins to factor a trinomial as shown below. Explain why the student is off to a bad start. 3x2 – 5x – 2 = (3x -‐ ____) (x -‐ ____ ) 4. Two students factor 2x2+ 20x + 42 and get two different answers: Student 1: (2x + 6) (x + 7) Student 2: (x + 3)(2x + 14) Do both answer check? Why don’t they agree? Is either answer completely correct? Explain 5. Explain the error in factorization shown below: a. X2-‐ 100 = (X – 50)(X + 50) b. X2-‐ 625 = (X2+ 25)(X2 – 25) DMA 060 WORKSHEET 4 QUADRATIC EQUATIONS AND GRAPHS Solve the equation. 1) (x - 2)(x + 4) = 0 2) (9y + 17)(4y + 29) = 0 3) x(5x + 25) = 0 4) x 2 + 5x - 14 = 0 5) x 2- 9x = -14 6) 16x
2 = 9 7) x(4x + 22) = 12 Represent the given condition using a single variable, x. 8) The length and width of a rectangle whose width is ten times its length. 9) Three consecutive even integers 10) The base and height of a triangle whose height is two less than eight times its base. Solve. 11) An object is thrown upward from the top of a 160
ft. building with an initial velocity of 48 feet per second. The height h of the object after t seconds is given by the quadratic equation ℎ = −16! ! + 48! + 160. When will the object hit the ground? 12) Two vehicles leave the same corner at the same time and travel at a right angle to each other. One vehicle travels 4 kilometers an hour faster than the other. If after one hour the vehicles are 20 kilometers apart, find the rate of each vehicle. Graphs!! Domain, Range, Open Up, Open Down, Max, Min?
Study Guide DMA 060 Name ____________________________ Date ____________________________ I.
Simplify each expression. Do not use negative exponents in the answer. 1. ! ! ! ! ∙ ! ! ! ! 2. (4!! ! ! ! )! 3.
4.
5.
II.
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Polynomials 6. Identify 2! ! + 8! − 12 as a monomial, binomial, or trinomial. Find the degree of each term and the degree of the polynomial. 7. Add or subtract the polynomials a. (15ℎ − 7ℎ + 12) + (3ℎ − 4ℎ + 10) !
2!! ! ! !
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b. Subtract 3! ! ! − 6!" + 11 from 2! ! − 5!" 8. Find a polynomial that represents the perimeter of the rectangle. (a – 4) in (5a2 + 2a – 8) in. 9. The cost C for an ice cream company to produce x pints of premium chocolate ice cream is given by the formula C = 3.25x + 600. If the company sells each pint for $4.25, the revenue R from their sale is given by the formula R = 4.25x. a. Write a formula that finds the profit P by subtracting cost from revenue when x pints of chocolate ice cream are sold. b. Will the company make a profit if they sell 500 pints of chocolate ice cream? Explain why or why not. III.
Multiply 10. 6! ! 2! ! − 3! + 4 11. ! − 8 2! + 7 12. (6! − 5!)! 13. (8! − 3)(8! + 3) 14. (3! − 2)(! ! − 4! + 2) 15. Find the area of the rectangle (2x -‐ 3) cm (4x -‐ 5) cm IV.
Divide !! ! !!!"!! !
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17. What binomial when multiplied by 3! ! , yields 21! ! − 15! ! ? V.
Factor Completely. If an expression cannot be factored, write “prime” 18. 27!! ! − 9!! ! + 45!! ! ! 19. ! ! − 4! − 5 20. 3!" + ! + 3!" + ! 21. 25! ! − 40! + 16 22. 2!! − 200! ! 23. The area of the rectangle is 25! ! − 40! + 16 !" ! . Find the dimensions of the rectangle. 24. Solve the equation: ! ! − 8! = −12 25. A window washer accidently drops a bucket from the top of a 144-‐foot building. The height h of the bucket after t seconds is given by ℎ = −16! ! + 144. When will the bucket hit the ground?