Unit 5 Quadratic Expressions and Equations 1/9/2017 2/8/2017 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
Table of Contents Vocabulary Organizer... 3 Adding Polynomials... 4 Standard Form... 4 Method 1: Horizontal Method... 4 Method 2: Algebra Tiles... 4 Subtracting Polynomials... 5 Method 1: Horizontal Method... 5 Method 2: Algebra Tiles... 5 Multiplying Polynomials by Monomials... 6 Method 1: Distributive Property... 6 Method 2: Box Method... 6 Multiplying Polynomials... 7 Method 1: Algebra Tiles... 7 Method 2: Box Method... 7 Method 3: Distributive Property... 8 Special Products... 9 Factoring with a GCF... 11 Factoring Trinomials (leading coefficient 1)... 14 Method 1: Algebra Tiles... 14 Method 2: Reverse Box Method... 14 Method 3: MA Table... 15 Factoring Trinomials (leading coefficient a)... 17 Difference of Squares... 20 Perfect Squares... 22
Vocabulary Organizer Do Now: With your group, think of as many words as you can that begin with the following prefixes: Mono- Bi- Tri- Poly- Term Polynomial Degree Leading Coefficient Quadratic Monomial Binomial Trinomial Perfect square trinomial
Adding Polynomials Standard Form *Note: All answers need to be written in standard form terms are in order from greatest to least degree. Example 1: Write 3x 2 + 4x 5 7x in standard form. Example 2: Write 5y 9 2y 4 6y 3 in standard form. Method 1: Horizontal Method Directions: Add like terms by grouping them horizontally. 1. (2x 2 + 5x 7) + (3 4x 2 + 6x) 2. (3y + y 3 5) + (4y 2 4y + 2y 3 + 8) Method 2: Algebra Tiles Directions: Create the polynomials using algebra tiles. Combine like terms. Remember that opposite pairs cancel. Free online tiles: goo.gl/omf9we (Note: You need Flash!) Click Manipulatives >> Algebra Tiles 1. (2x 2 + 5x 7) + (3 4x 2 + 6x)
Subtracting Polynomials Be Careful! Make sure to change ALL signs in the second polynomial! Method 1: Horizontal Method 1. (3 2x + 2x 2 ) (4x 5 + 3x 2 ) 2. (7p + 4p 3 8) (3p 2 + 2 9p) Method 2: Algebra Tiles 1. (3 2x + 2x 2 ) (4x 5 + 3x 2 ) Do Now: Add and subtract the following polynomials using your algebra tiles. 1. (4x 2 + 3x + 12) (3x 2 2x 1) 2. (2x 2 + 7x + 2) + (x 2 3x 6)
Multiplying Polynomials by Monomials 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
Multiplying Polynomials Warm Up: Arrange your algebra tiles on your product mat to create a rectangle with area x 2 + 7x + 12. What are the dimensions of your rectangle (length and width)? Method 1: Algebra Tiles 1. Find (x + 2)(x + 3). Sketch below. 2. Find (x + 1)(x + 5). 3. Find (x 2)(x 5). 4. Find (x 4)(2x + 3). Method 2: 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 3: Distributive Property 1. (x + 2)(x + 3) 2. (x 4)(2x + 3) 3. (2y 7)(3y + 5) 4. (6x + 5)(2x 2 3x 5) The Pool Problem A contractor is building a deck around a rectangular swimming pool. The deck is x feet from every side of the pool. Write an expression for the total area of the pool and deck.
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
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)
Factoring with a GCF Remember: GCF means greatest common factor. Do Now: 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? Using Algebra Tiles 1. Use algebra tiles to represent the polynomial 2x 8. 2. Arrange the tiles into a rectangle. The area of the rectangle represents the product, and its length and width represent the factors. What are the length and width (the factors)? 1. Use algebra tiles to represent the polynomial x 2 + 3x. 2. Arrange the tiles into a rectangle. What are the factors of x 2 + 3x? Use algebra tiles to factor each binomial. 1. 4x + 12 = 2. 4x 6 = 3. 3x 2 + 4x = 4. 10 2x = Where is the GCF shown in your algebra tile models?
Reverse Box Method Very similar to algebra tiles! 1. 27x 9 2. 2x 2 + 5x 3. 9m 4 18 4. 2x 5 + 3x 2 4x 5. 48u 5 72u 4 + 36u 3 Reverse Distributive Property 1. 27y 2 + 18y 2. 4a 2 b 8ab 2 + 2ab Steps: 1. Find the. 2. Take it. 3. What is left?. 3. 15w 3v 4. 7u 2 t 2 + 21ut 2 ut 5. 2x 2 y 2 + 5xy 3
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.
Factoring Trinomials (leading coefficient 1) Method 1: Algebra Tiles 1. Model x 2 + 4x + 3. 2. Arrange your tiles into a rectangle. 3. What are the factors? 1. Model x 2 + 8x + 12. 2. Arrange your tiles into a rectangle. This will take some trial and error. There is more than one correct way to do this! 3. What are your factors? 1. Model x 2 5x + 6. 2. Arrange your tiles into a rectangle. 3. What are the factors? 1. Model x 2 4x 5. 2. Arrange your tiles into a rectangle. 3. Remember: You can add zero pairs without changing the value of the polynomial. What are the factors? Method 2: Reverse Box Method Note: Again, this method is very similar to using algebra tiles. 1. You can fill in the top left and bottom right boxes right away. 2. Think: what needs to go in the other boxes? 1. Factor x 2 + 9x + 20. 2. Factor x 2 + 11x + 24.
Method 3: MA Table ax 2 + bb + c 1. Make a MA table (Multiply, Add). 2. You need two numbers that multiply to equal, and add to equal. 3. Write your answer in factored form. 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!
Solving Equations 1. Get everything on one side. 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.
Factoring Trinomials (leading coefficient a) Using Algebra Tiles 1. Factor 2x 2 + 5x + 3 using algebra tiles. 2. Factor 2x 2 + 9x + 5. 3. Factor 5x 2 13x + 6. Using a MA Table We can create a MA table, but the rules are slightly different. This time, we need two numbers that Multiply to and Add to. 1. Factor 3x 2 17x + 20. 5. Factor 2x 2 + 3x 5. 2. Factor 12x 2 + 11x 5. 6. Factor 14x 2 11x + 2. 3. Factor 5x 2 + 27x + 10. 7. Factor 3x 2 11x 20. 4. Factor 6x 2 + 17x + 5. 8. Factor 6x 2 x 15.
Tip!: Whenever there is a GCF, factor it out! It is always easier to work with smaller numbers. 1. Factor 4x 2 + 22x + 27. 3. Factor 5x 2 + 20x 105. 2. Factor 12x 2 69x + 45. 4. Factor 8x 2 4x 4. 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.
Solving Equations Factor, then use the ZPP! 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. Ken 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. Ben dives from a 36-foot platform. The equation h = 16t 2 + 14t + 36 models the dive. How long will it take Ben to reach the water?
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
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?
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
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.