Lesson 13: Exploring Factored Form
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- Blaze Horton
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1 Opening Activity Below is a graph of the equation y = 6(x 3)(x + 2). It is also the graph of: y = 3(2x 6)(x + 2) y = 2(3x 9)(x + 2) y = 2(x 3)(3x + 6) y = 3(x 3)(2x + 4) y = (3x 9)(2x + 4) y = (2x 6)(3x + 6) 1. Discussion How is this possible? What other equations can you write that are equivalent to these? Unit 11: More with Quadratics Factored Form S.117
2 2. We saw from the Opening Exercise that there are many ways to write equivalent equations or expressions. For the five expressions below, match each to the equivalent expressions. A. B. C. D. E. 4x x 2 9x 10x x x 8x 24 Matches: Matches: Matches: Matches: Matches: 1. 5( 2x + 2) 2. 3x( x 3) 3. 5x( x+ 5) 4. 10( x + 1) 2 ( x x) ( x 3) 13. 2( 5x + 5) 5. 4( 2x 6) 8. x( 5x+ 25) 11. x( 3x 9) 2 ( x + x) ( x 2 ) ( 4x 12) ( x 2 ) Discussion Which of the matched expressions do you think is the simplest? Circle your choices. Explain your thinking. Unit 11: More with Quadratics Factored Form S.118
3 In expressions 2, 3, 4, 6 and 10 the greatest common factor or GCF was factored out. For each problem below, find the GCF of the expression x x x 6. 22x x 2 15x 8. 10x x 3 Algebra tiles are often used to show how expressions can be factored. Below are the three tiles we ll be using. Each has been marked with their algebraic expression. 10. The 1-tile is 1 unit by 1 unit as shown below. Discuss how the dimensions of the other two tiles are represented by their algebraic expressions. Unit 11: More with Quadratics Factored Form S.119
4 11. The tiles for 2x 2 + 7x + 3 are shown below. Where do you see the factors of (2x + 1) and (x + 3)? 12. The model below is illustrating how x 2 + 5x + 6 can be factored. How is the model showing the factors of x 2 + 5x + 6? What are the factors? 13. Which of the following is the correct model for 2x 2 + 9x + 4? What are the factors of 2x 2 + 9x + 4? A. B. C. Unit 11: More with Quadratics Factored Form S.120
5 14. Let s look for some patterns between the standard form and the factored form. The first one has been started for you. Model Standard Form (sum of terms) Factored Form (product of terms) Patterns I Noticed y = 2x x + 8 y = (2x + 2)(x + 4) Unit 11: More with Quadratics Factored Form S.121
6 15. Draw algebra tile models for each expression below. Then determine the factors of each one. y = x 2 + 8x + 15 y = 3x 2 + 5x + 2 We can find the factors without having to draw out the algebra tiles each time by using a generic rectangle. For example, we can take the expression x 2 + 8x + 15 and place the first and last terms in a rectangle as shown below. Now we need the areas of these two rectangles to add to 8x, but have a product of Fill in the empty boxes of the generic rectangle above. What are the factors of x 2 + 8x + 15? How did this compare to your algebra tile model in Exercise 12? 17. Go back to Exercise 14 and draw in a generic rectangle for each algebra tile model. What additional patterns do you see? Unit 11: More with Quadratics Factored Form S.122
7 18. Create a generic rectangle for y = 3x 2 + 5x 2 and list the factors. 3x Create a generic rectangle for y = 2x 2 5x + 3 and list the factors. 2x Create a generic rectangle for y = 6x 2 x 1 and list the factors. 6x Create a generic rectangle for y = 4x 2 12x + 9 and list the factors. 4x 2 9 Unit 11: More with Quadratics Factored Form S.123
8 Lesson Summary Algebra Tile Model Example: y = x 2 + 5x + 6 = (x + 2)(x + 3) Generic Rectangle Example: y = x 2 + 5x + 6 = (x + 2)(x + 3) x 3 x x 2 3x 2 2x 6 Unit 11: More with Quadratics Factored Form S.124
9 Homework Problem Set Factor each equation in Problems 1 6. You may use either the algebra tile model or the generic rectangle. 1. y = x 2 + 2x y = x 2 + 4x y = x 2 + 6x y = x 2 + 7x y = x 2 + 6x y = x 2 + 8x + 16 Unit 11: More with Quadratics Factored Form S.125
10 Factor each equation in Problems y = 2x 2 + 7x y = 3x 2 + 4x y = 4x x y = 5x x y = 3x x y = 2x x + 15 Unit 11: More with Quadratics Factored Form S.126
11 Factor each equation in Problems y = 2x 2 7x y = 3x 2 14x y = 4x 2 + x y = 4x 2 4x y = 6x 2 + x y = 8x 2 10x 3 Unit 11: More with Quadratics Factored Form S.127
12 CHALLENGE PROBLEMS 19. Fill in the boxes with any numbers that make the equation true. 20. Fill in the blanks by finding the largest and smallest integers that will make the quadratic expression factorable. 21. An algebra tile model was started but not completed and it is missing some pieces. What trinomial could this be representing? What is it in factored form? Unit 11: More with Quadratics Factored Form S.128
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