Unit 5 AB 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 The Box Method... 6 Multiplying Polynomials... 7 Method 1: Algebra Tiles... 7 Method 2: Box Method... 7 Special Products... 9 Factoring with a GCF... 12 Factoring Trinomials (leading coefficient 1)... 15 Method 1: Reverse Box Method with MA Table... 15 Method 2: MA Table only... 16 Factoring Trinomials (leading coefficient a)... 18 Difference of Squares... 21 Perfect Squares... 23
Vocabulary Organizer Do Now: 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 in standard form. Example 2: Write in standard form. Method 1: Horizontal Method Directions: Add like terms by grouping them horizontally. 1. ( ) ( ) 2. ( ) ( ) 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. ( ) ( )
Subtracting Polynomials Be Careful! Make sure to change ALL signs in the second polynomial! Method 1: Horizontal Method 1. ( ) ( ) 2. ( ) ( ) Method 2: Algebra Tiles 1. ( ) ( )
Multiplying Polynomials by Monomials Do Now: Add and subtract the following polynomials using your algebra tiles. 1. ( ) ( ) 2. ( ) ( ) The Box Method 1. ( ) 2. ( ) 3. ( ) 4. ( ) 5. ( ) 6. ( )
Multiplying Polynomials Warm Up: Arrange your algebra tiles on your product mat to create a rectangle with area. What are the dimensions of your rectangle (length and width)? Method 1: Algebra Tiles 1. Find ( )( ). Sketch below. 2. Find ( )( ). 3. Find ( )( ). 4. Find ( )( ). Method 2: Box Method 1. ( )( ) 2. ( )( ) 3. ( )( ) 4. ( )( )
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 Squares of Sums We will make 3 squares with our algebra tiles. Write down the side length, area formula, and area. 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. ( ) 3. ( ) 2. ( ) 4. ( )
Squares of Differences What happens when some of our tiles are negative? We will swap all the tiles in our squares for red tiles. How does this affect the area? 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. ( ) 3. ( ) 2. ( ) 4. ( )
Product of a Sum and a Difference What happens when we only switch to negative tiles on one side? We will change the red tiles on one side back into positive tiles. How does this affect the area? 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. ( )( ) 3. ( )( ) 2. ( )( ) 4. ( )( )
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 and? 5. What is the GCF of and? Reverse Box Method 1. Steps: 1. Find the. 2. Put it. 3. What goes on top? 2.. 4. Write your final answer in. 3. 4.
More Reverse Box Method Practice 1. 2. 3. 4. 5. 6.
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, then. Examples (Already Factored): 1. Solve ( )( ). 2. Solve ( )( ). 3. Solve ( ). 4. Solve ( )( ). Examples (You need to do the factoring): 1. Solve. 2. Solve. 3. Solve. 4. Solve.
Factoring Trinomials (leading coefficient 1) ax bx c Method 1: Reverse Box Method with MA Table 1. In the top left square, write. 2. In the bottom right square, write. 3. Make a MA Table (Multiply, Add). 4. You need two numbers that multiply to equal, and add to equal. 5. Fill in the rest of the box. 6. Write your answer in factored form. 1. 2. 3. 4. 5. Factor. 6. Factor.
Method 2: MA Table only ax bx 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. 5. Factor. 2. Factor. 6. Factor. 3. Factor. 7. Factor 4. Factor. 8. Factor. 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. 4. Solve. 2. Solve 5. Solve. 3. Solve. 6. Solve.
Factoring Trinomials (leading coefficient a) Using a MA Table ax bx c We can create a MA table, but the rules are slightly different. This time, we need two numbers that Multiply to and Add to. Then, when we write the factors, we divide by a. Simplify as much as possible, then slide the denominator to the left. 1. Factor. 5. Factor. 2. Factor. 6. Factor. 3. Factor. 7. Factor. 4. Factor. 8. Factor.
Tip!: Whenever there is a GCF, factor it out! It is always easier to work with smaller numbers. 1. Factor. 3. Factor. 2. Factor. 4. Factor. 5. Factor. 6. Factor
Solving Equations Factor, then use the ZPP! 1. Solve. 2. Solve. 3. Solve 4. Solve. 5. Ken throws the discus at a school meet. The equation models his throw. After how many seconds does the discus hit the ground? 6. Ben dives from a 36-foot platform. The equation 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 ( )( )? Warm Up: Multiply: 1. ( )( ). 2. ( )( ). How can we use this trick to help us when factoring polynomials of the form? Summary: Factor each polynomial. 1. 4. 2. 5. 3. 6. 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. 10. 8. 11. 9. 12.
Solving Equations Factor as much as you can, then use ZPP. 1. Solve. 2. Solve. 3. Solve. 4. Solve. 5. Solve.
Perfect Squares Think Back: What is our shortcut for multiplying binomials of the form ( ) and ( )? Warm Up: Multiply: 1. ( )( ). 2. ( )( ). How can we use this trick to help us when factoring polynomials of the form? 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 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. 2. 3. 4. 5. 6.
Solving Equations with Perfect Squares 1. Factor. 2. Write perfect squares as repeated factors. 3. Use ZPP to solve! Solve each equation. 1. 2. Equations Using the Square Root Property Square Root Property: To solve a quadratic equation with the form of each side., take the 1. 2. ( ) 3. ( ) 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 can be used to approximate the number of seconds (t) it takes for the ball to reach height (h) from an initial height ( ). Find the time it takes the ball to reach the ground.