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ALGEBRA I
Hart Interactive – Algebra 1 M4 Lesson 16
Lesson 16: More Factoring Strategies
Opening Exploration – A Special Case
1. Consuela ran across the quadratic equation y = 4x2 – 16 and wondered how it could be factored. She
rewrote it as y = 4x2 + 0x – 16.
A. Use one of the methods you’ve learned to factor this quadratic function. B. What are the key features of the parabola’s graph?
x-intercepts: _____, _____ y-intercept: _____
vertex: (_____, _____)
C. Graph the quadratic in the grid at the right.
-20-19-18-17-16-15-14-13-12-11-10
-9-8-7-6-5-4-3-2-10123456789
1011121314151617181920
-3 -2 -1 0 1 2 3
Lesson 16: More Factoring Strategies Unit 9: More with Quadratics – Factored Form
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This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG I-M4-TE-1.3.0-09.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
ALGEBRA I
Hart Interactive – Algebra 1 M4 Lesson 16
Factor the following quadratic functions. Use Consuela’s idea of a 0 middle term if necessary. Look for patterns as you go. 2. y = x2 – 9 3. y = x2 – 25
4. What is the generic rule for factoring a quadratic in the form a2 – b2?
5. The expression a2 – b2 is called The Difference of Squares. Discuss with your partner where that name comes from.
On of the other special cases is Perfect Square Trinomials. You’ve already encountered a few of these but we’ll focus on them now and see why they are special.
6. Factor each of the following. Use any method.
A. y = x2 + 6x + 9 B. y = x2 – 4x + 4
7. A. What are the x-intercepts for these two equations? Remember the x-intercept is where y = 0.
B. What is different about the x-intercepts for these two equations?
Lesson 16: More Factoring Strategies Unit 9: More with Quadratics – Factored Form
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This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG I-M4-TE-1.3.0-09.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
ALGEBRA I
Hart Interactive – Algebra 1 M4 Lesson 16
8. Determine the key features of each quadratic and then graph the parabola.
A. y = x2 + 6x + 9 Standard Form y = (x +____)(x + ____) Factored Form
y = _______________ Vertex Form Key features: x-intercepts: _____, _____ y-intercept: _____ vertex: (_____, _____)
B. y = x2 – 4x + 4 Standard Form y = (x - ____)(x - ____) Factored Form
y = _______________ Vertex Form Key features: x-intercepts: _____, _____ y-intercept: _____ vertex: (_____, _____)
9. Both of the quadratic equations in Exercise 8 are perfect square trinomials. What is special about their graphs?
Lesson 16: More Factoring Strategies Unit 9: More with Quadratics – Factored Form
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This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG I-M4-TE-1.3.0-09.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
ALGEBRA I
Hart Interactive – Algebra 1 M4 Lesson 16
10. Let’s look at a couple of perfect square trinomials that have a ≠ 1. Factor each one.
A. y = 4x2 + 4x + 1 B. y = 9x2 – 12x + 4
11. Determine the key features of each quadratic and then graph the parabola.
A. y = 4x2 + 4x + 1 Standard Form y = (___x +____)(___x + ____) Factored Form
y = _______________ Vertex Form Key features: x-intercepts: _____, _____ y-intercept: _____ vertex: (_____, _____)
B. y = 9x2 – 12x + 4 Standard Form y =__________________ Factored Form
y = _______________ Vertex Form Key features: x-intercepts: _____, _____ y-intercept: _____ vertex: (_____, _____)
Lesson 16: More Factoring Strategies Unit 9: More with Quadratics – Factored Form
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This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG I-M4-TE-1.3.0-09.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
ALGEBRA I
Hart Interactive – Algebra 1 M4 Lesson 16
Practice Problems – Factor each perfect square trinomial or difference of squares. Notice that these are not written as quadratic functions. They are simply expressions – we could not graph them or tell their key features.
12. x2 + 4x + 4 13. x2 − 20x + 100 14. x2 + 2x + 1
15. x2 + 14x + 49 16. 64x2 + 48x + 9 17. 25x2 − 20x + 4
18. x2 − 8x + 16 19. x2 + 24x + 144 20. x2 − 4
21. 4x2 − 9 22. 25x2 − 36 23. x2 − 49
Lesson 16: More Factoring Strategies Unit 9: More with Quadratics – Factored Form
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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
ALGEBRA I
Hart Interactive – Algebra 1 M4 Lesson 16
Lesson Summary
Difference of Squares
(ax)2 − b2
(ax – b) (ax + b)
Perfect Square Trinomials
(ax)2 + 2abx + b2 (ax)2 – 2abx + b2
(ax + b)2 (ax – b)2
Lesson 16: More Factoring Strategies Unit 9: More with Quadratics – Factored Form
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This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG I-M4-TE-1.3.0-09.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
ALGEBRA I
Hart Interactive – Algebra 1 M4 Lesson 16
Homework Problem Set
Factor the following examples of the difference of perfect squares. Notice that these are not written as quadratic functions. They are simply expressions – we could not graph them or tell their key features.
1. 𝑡𝑡2 − 25 2. 4𝑥𝑥2 − 9
3. 16ℎ2 − 36𝑘𝑘2 4. 4 − 𝑏𝑏2
5. 𝑥𝑥4 − 4 6. 𝑥𝑥6 − 25
7. 9𝑦𝑦2 − 100𝑧𝑧2 8. 𝑎𝑎4 − 𝑏𝑏6
9. Challenge 𝑟𝑟4 − 16𝑠𝑠4 (Hint: This one factors twice.)
Lesson 16: More Factoring Strategies Unit 9: More with Quadratics – Factored Form
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This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG I-M4-TE-1.3.0-09.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
ALGEBRA I
Hart Interactive – Algebra 1 M4 Lesson 16
10. For each of the following, factor out the greatest common factor.
a. 6𝑦𝑦2 + 18 b. 27𝑦𝑦2 + 18𝑦𝑦 c. 21𝑏𝑏 − 15𝑎𝑎 d. 14𝑐𝑐2 + 2𝑐𝑐 e. 3𝑥𝑥2 − 27
11. The measure of a side of a square is 𝑥𝑥 units. A new square is formed with each side 6 units longer than the original square’s side. Write an expression to represent the area of the new square. (Hint: Draw the new square and count the squares and rectangles.)
Original Square 𝑥𝑥
Lesson 16: More Factoring Strategies Unit 9: More with Quadratics – Factored Form
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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
ALGEBRA I
Hart Interactive – Algebra 1 M4 Lesson 16
12. In the accompanying diagram, the width of the inner rectangle is represented by 𝑥𝑥 − 3 and the length by 𝑥𝑥 + 3. The width of the outer rectangle is represented by 3𝑥𝑥 − 4 and the length by 3𝑥𝑥 + 4. a. Write an expression to represent the area of the larger rectangle. b. Write an expression to represent the area of the smaller rectangle.
MIXED REVIEW
Factor completely.
13. 9𝑥𝑥2 − 25𝑥𝑥 14. 9𝑥𝑥2 − 25 15. 9𝑥𝑥2 − 30𝑥𝑥 + 25 16. 2𝑥𝑥2 + 7𝑥𝑥 + 6 17. 6𝑥𝑥2 + 7𝑥𝑥 + 2 18. 8𝑥𝑥2 + 20𝑥𝑥 + 8
19. 3𝑥𝑥2 + 10𝑥𝑥 + 7 20. 4𝑥𝑥2 + 4𝑥𝑥 + 1
Lesson 16: More Factoring Strategies Unit 9: More with Quadratics – Factored Form
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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
ALGEBRA I
Hart Interactive – Algebra 1 M4 Lesson 16
CHALLENGE PROBLEMS 21. The area of the rectangle at the right is represented by the expression
𝟏𝟏𝟏𝟏𝒙𝒙𝟐𝟐 + 𝟏𝟏𝟐𝟐𝒙𝒙 + 𝟐𝟐 square units. Write two expressions to represent the dimensions, if the length is known to be twice the width.
22. Two mathematicians are neighbors. Each owns a separate rectangular plot of land that shares a boundary and has the same dimensions. They agree that each has an area of 2𝑥𝑥2 + 3𝑥𝑥 + 1 square units. One mathematician sells his plot to the other. The other wants to put a fence around the perimeter of his new combined plot of land. How many linear units of fencing does he need? Write your answer as an expression in 𝑥𝑥.
Note: This question has two correct approaches and two different correct solutions. Can you find them both?
𝟏𝟏𝟏𝟏𝒙𝒙𝟐𝟐 + 𝟏𝟏𝟐𝟐𝒙𝒙 + 𝟐𝟐
Lesson 16: More Factoring Strategies Unit 9: More with Quadratics – Factored Form
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This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG I-M4-TE-1.3.0-09.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
ALGEBRA I
Hart Interactive – Algebra 1 M4 Lesson 16
Spiral Review – Factoring Lessons 13 – 15
Factor the following quadratic expressions.
23. 2𝑥𝑥2 + 10𝑥𝑥 + 12 24. 6𝑥𝑥2 + 5𝑥𝑥 − 6
25. x2 – 12x + 20 26. x2 – 21x – 22 27. 2𝑥𝑥2 − 𝑥𝑥 − 10 28. 6𝑥𝑥2 + 7𝑥𝑥 − 20 29. x2 – 2x – 15 30. x2 + 2x – 15 31. 4x2 + 12x + 9 32. 49x2 + 28x + 4
Lesson 16: More Factoring Strategies Unit 9: More with Quadratics – Factored Form
S.157
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG I-M4-TE-1.3.0-09.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
ALGEBRA I
Hart Interactive – Algebra 1 M4 Lesson 16
Lesson 16: More Factoring Strategies Unit 9: More with Quadratics – Factored Form
S.158
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG I-M4-TE-1.3.0-09.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.