A Resource Aligned with New York State Algebra 1 Module 4
Table of Contents
Multiplying Polynomial Expressions ………………………………………………………………………………………………… 5
Factoring GCF and Difference of Squares ………………………………………………………………………………………… 12
Factoring Trinomials ……………………………………………………………………………………………………………………….. 18
Factoring Quadratic Expressions with a Leading Coefficient ……………………………………………………………. 24
The Zero Product Property …………………………………………………………………………………………………………….. 29
Quadratic Equations ………………………………………………………………………………………………………………………. 35
Creating and Solving Quadratic Equations ……………………………………………………………………………….…….. 41
Exploring the Symmetry in Graphs of Quadratic Functions ……………………………………………………….……. 48
Graphing Quadratic Functions from Factored Form ………………………………………………………………….……. 55
Interpreting Quadratic Functions from Graphs and Tables …………………………………………………………….. 64
Completing the Square ……………………………………………………………………………………………………………….….. 70
Completing the Square with Leading Coefficients …………………………………………………………………………… 76
Solving Quadratics by Completing the Square …………………………………………………………………………….…… 80
Simplifying Radicals ………………………………………………………………………………………………………………………… 87
Quadratic Formula ……………………………………………………………………………………………………………………….…. 92
Nature of the Roots of Quadratic Equations ……………………………………………………………………………………. 98
Graphing Quadratic Functions from the Vertex Form ……………………………………………………………………. 104
Graphing Quadratic Functions from the Standard Form ……………………………………………………………… 109
Graphing Cubic, Square Root, and Cube Root Functions ……………………………………………………………….. 117
Interpreting Quadratic Functions and Their Graphs ……………………………………………………………………… 122
Key Features of Quadratic Graphs ……………………………………………………………………………………………….. 126
End of Module Review (Sample Regents Exam Questions) ………………………………………………………….. 130
Name ______Multiplying Polynomial Expressions
Show all your work! Opening Exercise
Write expressions for the areas of the two rectangles in the figures given below.
Now write an expression for the area of this rectangle:
Multiplying Polynomial Expressions
Learning Target: I can multiply polynomial expressions.
Example: Multiply Two Binomials
Fill in the table to identify the partial products of (x+2)(x+5). Then write the product of (x+2)(x+5) in standard form.
Multiply Two Binomials—Without the Aid of a Table
Multiply: (x+2)(x-5)
Example: The Difference of Squares
Find the product of (x+2)(x-2). Use the distributive property to distribute the first binomial over the second.
With the use of a table:
Without the use of a table:
Example: The Square of a Binomial
To square a binomial, like (x+3)2, multiply the binomial by itself.
Square the binomial.
A. (a+6)2
B. (5-w)2
C. (a+b)2
D. (a-b)2
Exercises: Multiply each pair of binomials
1. (x+1)(x-7)
2. x+9(x+2)
3. (x-5)(x-3)
4. x+152(x-1)
Name ______Multiplying Polynomial Expressions
Show all your work! Problem Set
1. Multiply:
a. (n-5)(n + 5)
b. (4-y)(4+y)
c. (k+10)2
d. (4+b)2
2. Find the dimensions:
The measure of a side of a square is x 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.
3. In the accompanying diagram, the width of the inner rectangle is represented by x-3 and its length by x+3. The width of the outer rectangle is represented by 3x+4 and its length by 3x-4.
a. Find the area of the larger rectangle.
b. Find the area of the smaller rectangle.
c. Express the area of the pink shaded region as a polynomial in terms of x. (Hint: You will have to add or subtract polynomials to get your final answer.)
Name ______Multiplying Polynomial Expressions
Show all your work! Exit Ticket
When you multiply two terms by two terms, you should get four terms. Why is the final result when you multiply two binomials sometimes only three terms? Give an example of how your final result can end up with only two terms.
Name ______Factoring GCF and Difference of Squares
Show all your work! Opening Exercise
The total area of this rectangle is represented by 3a2 + 3a. Find expressions for the dimensions of the total rectangle.
Factoring GCF and Difference of Squares
Learning Target: I can factor polynomial expressions by using a greatest common factor or difference of squares.
Factor each by factoring out the Greatest Common Factor:
1. 10ab + 5a
2. 3g3h – 9g2h+12h
3. 6x2y3+9xy4+18y5
4. Factor the following examples of the difference of perfect squares.
a. t2-25
b. 4x2-9
c. 16h2-36k2
d. 4-b2
e. x4-4
f. x6-25
g. a2-b2
Factor each of the following differences of squares completely:
5. 9y2-100z2
6. a4-b6
7. r4-16s4 (Hint: This one will factor twice.)
Name ______Factoring GCF and Difference of Squares
Show all your work! Problem Set
For each of the following factor out the greatest common factor:
a. 6y2+18
b. 27y2+18y
c. 21b-15a
d. 14c2+2c
e. 3x2-27
Name ______Factoring GCF and Difference of Squares
Show all your work! Exit Ticket
Factor completely: x6-81x2
Name ______Factoring Trinomials
Show all your work! Opening Exercises
1A. List the factor pairs of 24.
______x ______= 24 ______x ______= 24
______x ______= 24 ______x ______= 24
______x ______= 24 ______x ______= 24
______x ______= 24 ______x ______= 24
1B. Which factor pair of 24 adds up to – 11?
1C. Factor x2 – 11x + 24.
2A. List the factor pairs of –32.
2B. Which factor pair of –32 adds up to –14?
2C. Factor x2 – 14x – 32.
Factoring Trinomials
Learning Target: I can factor trinomial expressions and understand that factoring reverses multiplication.
Exercises
1. Use a table to aid in finding the product of (x+7)(x+3).
2. Factor: x2+10x+21
3. Factor: x2+8x+7
4. Factor: m2+m-90
5. Factor: k2-13k+40
6. Factor: -100+99v+v2
7. Factor Completely: 2x3-50x
8. Factor Completely: –16t2+32t+48
Name ______Factoring Trinomials
Show all your work! Problem Set
4. Factor these trinomial as the product of two binomials and check your answer by multiplying:
a. x2+3x+2
b. x2-8x+15
c. x2+8x+15
Factor completely:
d. 4m2-4n2
e. -2x3-2x2+112x
f. y8-81x4
2. The parking lot at Gene Simon’s Donut Palace is going to be enlarged, so that there will be an
additional 30 ft. of parking space in the front and 30 ft. on the side of the lot. Write an expression in
terms of x that can be used to represent the area of the new parking lot. Explain how your solution is
demonstrated in the area model.
Name ______Factoring Trinomials
Show all your work! Exit Ticket
Name ______Factoring Quadratic Expressions with a Leading Coefficient
Show all your work! Opening Exercises
1. Simplify: 4n(n2 +2x – 5)
2. Simplify: (x + 7)(x – 3)
3. Factor x2 + 5x + 6
4. Factor x2 + x – 6
5. Factor x2 – x – 6
6. Factor x2 – 5x + 6
Factoring Quadratic Expressions with a Leading Coefficient
Learning Target: I can factor quadratic expressions with a leading coefficient.
1. Multiply the binomials: (2x+3)(1x+5)
2. Factor the trinomial: 2x2+13x+15
3. Factor: 3x2-x-4
Exercises 1–6
Factor the expanded form of these quadratic expressions. Pay particular attention to the negative/positive signs.
1. 3x2-2x-8
2. 3x2+10x-8
3. 3x2+x-14
4. 2x2-21x-36
5. -2x2+3x+9
6. r2+64r+916
Name ______Factoring Quadratic Expressions with a Leading Coefficient
Show all your work! Problem Set
Factor the following quadratic expressions.
1. 3x2-2x-5
2. –2x2+5x-2
3. 5x2+19x-4
4. 4x2-12x+9 [This one is tricky, but look for a special pattern.]
5. 3x2-13x+12
Name ______Factoring Quadratic Expressions with a Leading Coefficient
Show all your work! Exit Ticket
Factor completely.
1. x2 + 10x – 24
2. x2 – 81
3. 2x2 + 5x + 2
4. 2x2 + 6x + 4
5. 6x2 – 9x
Name ______The Zero Product Property
Show all your work! Opening Exercise
Consider the equation a∙b∙c∙d=0. What values of a, b, c, and d would make the equation true?
The Zero Product Property
Learning Target: I can use the zero product property to solve equations.
Exercises 1–4
Find values of c and d that satisfy each of the following equations. (There may be more than one correct answer.)
1. cd=0
2. (c-5)d=2
3. (c-5)d=0
4. (c-5)(d+3)=0
Example 1
For each of the related questions below use what you know about the Zero-Product Property to find the answers.
a. The area of a rectangle can be represented by the expression, x2+2x-3. Write each dimension of this rectangle as a binomial, and then write the area in terms of the product of the two binomials.
b. Suppose the area of the rectangle is 21 square units. Rewrite the equation so that it is equal to zero and solve.
c. What are the actual dimensions of the rectangle?
d. If a smaller rectangle, which can fit inside the first rectangle, has an area that can be expressed by the equation x2-4x-5. What are the dimensions of the smaller rectangle?
e. What value for x would make the smaller rectangle have an area of 13 that of the larger?
Exercises 5–8
Solve. Show your work:
5. x2-11x+19= –5
6. 7x2+x=0
7. 7r2-14r= –7
8. 2d2+5d-12=0
Name ______The Zero Product Property
Show all your work! Problem Set
1. x2-11x+ 19= –5
2. 7x2+2x=0
3. b2+5b-35=3b
4. 7r2-14r= –7
Name ______The Zero Product Property
Show all your work! Exit Ticket
Name ______Quadratic Equations
Show all your work! Opening Exercise
Solve each quadratic equation.
1. y2+6y=0
2. a2-10a=24
3. x2=25
4. 5n2=45
5. (x-3)2=4
Quadratic Equations
Learning Target: I can solve quadratic equations.
Example 1
A physics teacher put a ball at the top of a ramp and let it roll down toward the floor. The class determined that the height of the ball could be represented by the equation, h= –16t2+4, where the height is measured in feet from the ground and time in seconds.
a. What do you notice about the structure of the quadratic expression in this problem? How can this structure help us when we apply this equation?
b. In the equation, explain what the 4 represents.
c. Use the equation to determine the time it takes the ball to reach the floor.
d. Now consider the two solutions for t. Which one is reasonable? Does the final answer make sense based on this context? Explain.
Exercises: Solve each equation.
4. 3x2-9=0
5. (x-3)2=1
6. 4(x-3)2=1
7. 2(x-3)2=12
8. Peter is a painter and he wonders if he would have time to catch a paint bucket dropped from his ladder before it hits the ground. He drops a bucket from the top of his 9-foot ladder. The height, h, of the bucket during its fall can be represented by the equation, h= –16t2+9, where the height is measured in feet from the ground, and the time since the bucket was dropped is measured in seconds. After how many seconds does the bucket hit the ground? Do you think he could catch the bucket before it hits?
Name ______Quadratic Equations
Show all your work! Problem Set
9. Factor completely: 15x2-40x-15
Solve:
10. 4x2=9
11. 3y2-8=13
12. Mischief is a Toy Poodle who competes with her trainer in the agility course. Within the course, Mischief must leap through a hoop. Mischief’s jump can be modeled by the equation h=-16t2+12t, where h is the height of the leap in feet and t is the time since the leap, in seconds. At what values of t does Mischief start and end the jump?
Name ______Quadratic Equations
Show all your work! Exit Ticket
Name ______Creating and Solving Quadratic Equations
Show all your work! Opening Exercise
The length of a rectangle is 5 in. more than twice a number. The width is 4 in. less than the same number. If the perimeter of the rectangle is 26 in., find the unknown number.
Creating and Solving Quadratic Equations
Learning Target: I can interpret word problems to create equations in one variable and solve them by factoring.
Example 1
The length of a rectangle is 5 in. more than twice a number. The width is 4 in. less than the same number. If the area of the rectangle is 15, find the unknown number.
Example 2
A picture has a height that is 43 its width. It is to be enlarged so that the ratio of height to width remains the same but the area is 192 sq in. What are the dimensions of the enlargement?
Exercises
Solve the following problems. Be sure to indicate if a solution is to be rejected based on the contextual situation.
1. The length of a rectangle 4 cm more than 3 times its width. If the area of the rectangle is 15 cm2, find the width.
2. The ratio of length to width in a rectangle is 2:3. Find the length of the rectangle when the area is 150 sq in.
3. Karen wants to plant a garden and surround it with decorative stones. She has enough stones to enclose a rectangular garden with a perimeter of 68 ft., and she wants the garden to cover 240 sq ft. What will the length and width of her garden be?
4. Find two consecutive odd integers whose product is 99. [Note: There are two different pairs of consecutive odd integers and only an algebraic solution will be accepted.]
Name ______Creating and Solving Quadratic Equations
Show all your work! Problem Set
1. The length of a rectangle is 2 cm less than its width. If the area of the rectangle is 35 cm2, find the width.
2. The ratio of length to width (measured in inches) in a rectangle is 4:7. Find the length of the rectangle if the area is known to be 700 sq in.
3. A student is painting an accent wall in his room where the length of the room is 3 ft. more than its width. The wall has an area of 130 ft2. What are the length and the width, in feet?
4. Find two consecutive even integers whose product is 80. (There are two pairs and only an algebraic solution will be accepted.)
Name ______Creating and Solving Quadratic Equations
Show all your work! Exit Ticket
1. The perimeter of a rectangle is 54 cm. If the length is 2 cm more than a number, and the width is 5 cm less than twice the same number, what is the number?
2. A plot of land for sale has a width of ft. and a length that is 8 ft. less than its width. A farmer will only purchase the land if it measures 240 ft2. What value for will cause the farmer to purchase the land?
Name ______Exploring the Symmetry in Graphs of Quadratic Functions
Show all your work! Opening Exercise
Learning Target: I can identify key features of the graphs of quadratic functions.
Use the graphs of quadratic functions A and B to fill in the table.