Unit 5 Quadratic Expressions and Equations Test Date: Name: By the end of this unit, you will be able to Add, subtract, and multiply polynomials Solve equations involving the products of monomials and polynomials Find squares of sums and differences Find the product of a sum and a difference Factor polynomials Solve quadratic equations Factor binomials that are the difference of squares Use the difference of squares to solve equations Factor perfect square trinomials Solve equations involving perfect squares
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Table of Contents Vocabulary Organizer... 4 5.1 Adding Polynomials... 5 Standard Form... 5 5.1 Subtracting Polynomials... 5 5.2 Multiplying Polynomials by Monomials... 6 Method 1: Distributive Property... 6 Method 2: Box Method... 6 5.2 Multiplying Polynomials by Polynomials... 7 Method 1: Box Method... 7 Method 2: Distributive Property... 7 5.3 Special Products... 8 Squares of Sums... 8 Squares of Differences... 9 Product of a Sum and a Difference... 9 5.4 Factoring with a GCF... 10 Reverse Box Method... 10 Reverse Distributive Property... 11 Solving Equations by Factoring... 12 5.5 Factoring Trinomials (leading coefficient 1)... 13 Diamond Problems and Factor Tables... 13 Factoring... 14 Solving Equations... 15 5.6 Factoring Trinomials (leading coefficient a)... 16 Slide and Divide... 16 Solving Equations... 18 5.7 Difference of Squares... 19 Solving Equations... 20 5.8 Perfect Squares... 21 Identifying Perfect Square Trinomials... 21 Solving Equations with Perfect Squares... 22 Equations Using the Square Root Property... 22 Interpreting Your Answer in Word Problems... 22 3
Vocabulary Organizer Warm Up: With your group, think of as many words as you can that begin with the following prefixes: Mono- Bi- Tri- Term Polynomial Degree Leading Coefficient Quadratic Monomial Binomial Trinomial Perfect square trinomial 4
5.1 Adding Polynomials Standard Form *Note: All answers need to be written in standard form terms are in order from. Example 1: Write 3x 2 + 4x 5 7x in standard form. Example 2: Write 5y 9 2y 4 6y 3 in standard form. Directions: Add like terms by grouping them. 1. (2x 2 + 5x 7) + (3 4x 2 + 6x) 2. (3y + y 3 5) + (4y 2 4y + 2y 3 + 8) 5.1 Subtracting Polynomials Be Careful! Make sure to change ALL signs in the second polynomial! 1. (3 2x + 2x 2 ) (4x 5 + 3x 2 ) 2. (7p + 4p 3 8) (3p 2 + 2 9p) 5
5.2 Multiplying Polynomials by Monomials Warm Up: Add or subtract the following polynomials. 1. (4x 2 + 3x + 12) (3x 2 2x 1) 2. (2x 2 + 7x + 2) + (x 2 3x 6) Method 1: Distributive Property 1. 3x 2 (7x 2 x + 4) 2. 5a 2 ( 4a 2 + 2a 7) 3. 6d 3 (3d 4 2d 3 d + 9) Method 2: Box Method 1. 3x 2 (7x 2 x + 4) 2. 5a 2 ( 4a 2 + 2a 7) 3. 6d 3 (3d 4 2d 3 d + 9) Solving Equations Multiplying monomials by polynomials helps us solve complicated equations. Example: 2x(x + 4) + 7 = (x + 8) + 2x(x + 1) + 12 6
5.2 Multiplying Polynomials by Polynomials Method 1: Box Method 1. (x + 2)(x + 3) 2. (x 4)(2x + 3) 3. (2x + 3)(x + 5) 4. (2y 2 + 3y 1)(3y 2 5y 2) Method 2: Distributive Property 1. (x + 2)(x + 3) 2. (x 4)(2x + 3) 3. (2y 7)(3y + 5) 4. (6x + 5)(2x 2 3x 5) Application: A contractor is building a deck around a rectangular swimming pool. The homeowner would like one side of the pool to be 6 feet longer than the other. Write an expression for the area of the pool. 7
5.3 Special Products Do Now: Create a square out of algebra tiles. (Remember: All sides should be the same length in a square) What is the side length of your square? What is the formula for the area of your square? What is the area of your square? (Multiply polynomials) *You may wish to sketch your square at right* Squares of Sums Write down the side length, area formula, and area of at least three other squares. Square #1 Square #2 Square #3 Side length: Side length: Side length: Area formula: Area formula: Area formula: Area: Area: Area: What do you notice? Summary: 1. (x + 3) 2 2. (3x + 5) 2 8
Squares of Differences What happens when some of our tiles are negative? Swap all the x tiles in your square for red x tiles. How does this affect the area? Summary: 1. (6p 1) 2 2. (x 2) 2 Product of a Sum and a Difference What happens when we only switch to negative tiles on one side? Change the red x tiles on one side back into positive x tiles. How does this affect the area? Summary: 1. (x + 4)(x 4) 2. (3x + 2)(3x 2) 9
5.4 Factoring with a GCF Remember: GCF means greatest common factor. Warm Up: 1. List all the factors of 48. 2. What is the GCF of 9 and 12? 3. What is the GCF of 65 and 39? 4. What is the GCF of x 3 and x 2? 5. What is the GCF of 4x 5 and 6x 3? Reverse Box Method 1. 27x 9 2. 2x 2 + 5x 3. 9m 4 18 4. 2x 5 + 3x 2 4x 5. 48u 5 72u 4 + 36u 3 10
Reverse Distributive Property 1. 27y 2 + 18y Steps: 1. Find the. 2. Take it. 3. What is left? 2. 4a 2 b 8ab 2 + 2ab. 3. 15w 3v 4. 7u 2 t 2 + 21ut 2 ut 5. 2x 2 y 2 + 5xy 3 11
Solving Equations by Factoring The Zero Product Property In words: If the product of two factors is 0, then at least one of the factors must be. In symbols: If a b = 0, then. Examples (Already Factored): 1. Solve (x 4)(x + 7) = 0. 2. Solve (2d + 6)(3d 15) = 0. 3. Solve 3n(n + 2) = 0. 4. Solve (x 2)(4x + 1) = 0. Examples (You need to do the factoring): 1. Solve c 2 = 3c. 2. Solve 8b 2 = 40b. 3. Solve x 2 = 10x. 4. Solve 3k 2 = 24k. 12
5.5 Factoring Trinomials (leading coefficient 1) Warm Up: List all the factors of the following numbers: 27 13 4 36 Diamond Problems and Factor Tables The rules of diamond problems: You must find 2 numbers to put on the left and right sides of the X. When you multiply your two numbers together, they must equal the top number. When you add your two numbers together, they must equal the bottom number. Good luck! 1. 2. 3. 4. 5. 6. Tips and Tricks: 13
Factoring 1. Draw a diamond problem. For ax 2 + bx + c, write a c at the top and b at the bottom. 2. Find two numbers that multiply to equal, and add to equal. a. If you can t find them, make a table of all the factors of. 3. Write your answer in factored form: (x + )(x + ). 1. Factor x 2 + 9x + 20. 5. Factor x 2 8x + 12. 2. Factor x 2 + 11x + 24. 6. Factor x 2 11x + 28. 3. Factor x 2 + 15x + 36. 7. Factor x 2 + 2x 15. 4. Factor x 2 + 10x + 9. 8. Factor x 2 7x 18. Tip!: You can always check your answer by multiplying your factors together! 14
Solving Equations 1. Make sure equation is equal to. 2.. 3. Set each factor equal to. 4. each equation. 1. Solve x 2 + 3x 18 = 0. 4. Solve x 2 3x = 70. 2. Solve x 2 15x + 36 = 0. 5. Solve x 2 + 12x = 32. 3. Solve x 2 3x + 2 = 0. 6. Solve x 2 x 72 = 0. 15
5.6 Factoring Trinomials (leading coefficient a) Slide and Divide 1. If possible, factor out a. 2. Draw a diamond problem. For ax 2 + bx + c, write a c at the top and b at the bottom. 3. Find two numbers that multiply to equal, and add to equal. a. If you can t find them, make a table of all the factors of. 4. Write your answer in factored form: (x + )(x + ). Divide by! 5. If possible, reduce/simplify any fractions. 6. Kick any denominators to the front of the factor. 1. Factor 3x 2 17x + 20. 4. Factor 6x 2 + 17x + 5. 2. Factor 12x 2 + 11x 5. 5. Factor 2x 2 + 3x 5. 3. Factor 5x 2 + 27x + 10. 6. Factor 14x 2 11x + 2. 16
7. Factor 3x 2 11x 20. 8. Factor 6x 2 x 15. Don t forget step 1! Whenever there is a GCF, factor it out! Slide and divide doesn t work if you don t! 1. Factor 12x 2 69x + 45. 3. Factor 8x 2 4x 4. 2. Factor 5x 2 + 20x 105. Challenge Problems 1. Six times the square of a number x plus 11 times the number equals 2. What are the possible values for x? 2. Factor 4x 2 15x 14. 3. Factor 4x 2 25. 17
Solving Equations Factor, then use the ZPP! Shortcut: 1. Solve 2x 2 + 9x 18 = 0. 2. Solve 7x 2 + 3 = 10x. 3. Solve 2x 2 13x = 15. 4. Solve 3x 2 5x = 2. 5. Alex throws the discus at a school meet. The equation h = 16t 2 + 38t + 5 models his throw. After how many seconds does the discus hit the ground? 6. Hannah dives from a 36-foot platform. The equation h = 16t 2 + 14t + 36 models the dive. How long will it take Hannah to reach the water? 18
5.7 Difference of Squares Think Back: What is our shortcut for multiplying binomials of the form (a + b)(a b)? Warm Up: Multiply: 1. (x + 5)(x 5) =. 2. (2x + 3)(2x 3) =. How can we use this trick to help us when factoring polynomials of the form a 2 b 2? Summary: Factor each polynomial. 1. x 2 64 4. 121 4b 2 2. x 2 26 5. 81 c 2 3. 16h 2 9a 2 6. 64g 2 h 2 Tip!: Don t forget to factor out a GCF whenever you can. Tip #2!: We can use this technique more than once, or mix this technique with other techniques. Factor each polynomial. 7. 27g 3 3g 10. b 4 16 8. 9x 3 4x 11. 625 x 4 9. 4y 3 + 9y 12. 5x 5 45x 19
Solving Equations Factor as much as you can, then use ZPP. 1. Solve 9m 2 144 = 0. 2. Solve 18x 3 = 50x. 3. During an accident, skid marks may result from sudden braking. The formula 1 24 s2 = d approximates a vehicle s speed s in miles per hour given the length d in feet of the skid mark on dry concrete. If a skid mark is 54 feet long, how fast was the car traveling when the brakes were applied? 20
5.8 Perfect Squares Think Back: What is our shortcut for multiplying binomials of the form (a + b) 2 and (a b) 2? Warm Up: Multiply: 1. (x + 5)(x + 5) =. 2. (2x 3)(2x 3) =. How can we use this trick to help us when factoring polynomials of the form a 2 + 2ab + b 2? Summary: Identifying Perfect Square Trinomials Before we can use this shortcut to help us factor, we need to know how to recognize perfect square trinomials. We know that they must fit the form a 2 + 2ab + b 2. Questions to ask yourself: 1. Is the first term a perfect square? 2. Is the last term a perfect square? 3. Take the square root of the 1 st and 3 rd terms to find a and b. Is the middle term equal to 2ab? Determine whether each trinomial is a perfect square. Write yes or no. If so, factor it. 1. 4y 2 + 12y + 9 2. 9x 2 6x + 4 3. 9y 2 + 24y + 16 4. 2a 2 + 10a + 25 5. 25x 2 30x + 9 6. 49y 2 + 42y + 36 21
Solving Equations with Perfect Squares 1. Factor. 2. Write perfect squares as repeated factors. 3. Use ZPP to solve! Solve each equation. 1. 9x 2 48x = 64 2. a 2 + 12a + 36 = 0 Equations Using the Square Root Property Square Root Property: To solve a quadratic equation with the form x 2 = n, take the of each side. 1. x 2 = 25 2. (y 6) 2 = 81 3. (x + 6) 2 = 12 Interpreting Your Answer in Word Problems Some word problems should only have one answer. You must examine the problem to determine which answer is best. Example: During an experiment, a ball is dropped from a height of 205 feet. The formula h = 16t 2 + h 0 can be used to approximate the number of seconds (t) it takes for the ball to reach height (h) from an initial height (h 0 ). Find the time it takes the ball to reach the ground. 22