Chapter 10 Using Factoring
Section 1 – Factors and Greatest Common Factors
Essential Question: Many situations in real-life can be represented by quadratic equations.
How can these quadratic equations be solved using factoring to determine specific values in the
given situation?
Objective: Compile the factorization of monomials; Compute the greatest common factor of
monomials
Bell-ringer: Follow the direction below to complete the anticipation chart.
 Read each statement.
 Decide whether you Agree (A) or Disagree (D) with the statement.
 Write A or D in the first column OR if you are not sure whether you agree or disagree,
write NS (Not Sure).
STEP 1
Statement
STEP 2
A, D, or
A or D
NS
1. The number 1 is a prime number because its only factors are 1 and
itself.
2. A composite number is any whole number greater than 1 that has
more than two factors.
3. The prime factorization of a number is the product of two prime
numbers that equal that number.
4. Any two numbers that have a greatest common factor of 1 are said to
be relatively prime.
5. If the product of any two factors is 0, then at least one of the factors
must equal 0.
6. A quadratic trinomial has a degree of 4.
7. To solve an equation such as x2 = 8 + 2x, take the square root of both
sides.
8. The polynomial 3s2 – s – 2 cannot be factored because the coefficient
of s2 is not 1.
9. The polynomial t2 + 16 is not factorable.
10. The numbers 16, 64, and 121 are perfect squares.
Factoring:
Prime Numbers:
Composite Numbers:
Prime Factorization:
Practice - Find the prime factorization of each of the following.
1. 140
2. 84
3. -210
4. -448
Factored Form:
Practice – Express each monomial in its factored form.
5. 45x3y2
6. 48a2b2
7. 77ab2
8. 4b3d2
Greatest Common Factor (GCF):
Practice – Find the greatest common factor of each group of monomials.
9. 54, 63, and 180
10. 64, 80, and 96
11. 40a2b and 48ab4
12. 12a2b and 90a2b2c
Where might we use this?
13. Cookies A bakery packages its fat-free cookies in two sizes of boxes. One box contains 18
cookies, and the other contains 24 cookies. To keep the cookies fresh, the bakery plans to
wrap a smaller number of cookies in cellophane before they are placed in the boxes. To save
money, the bakery wants to use the same size cellophane packages for each box and to place
the greatest possible number of cookies in each cellophane package.
a. How many cookies should the bakery place in each cellophane package?
b. How many cellophane packages will go in each size of box?
Ticket-Out: (To be completed in composition book.)
1. Factor 145ab2c3 completely.
2. Find the GCF of 24fg5 and 56f3g.
Independent Practice
Factor each monomial completely.
1. -70
2. 66z2
3. -102x3y
Find the GCF of the given monomials.
4. -34, 51
5. 12an2, 40a2
6. 12a2b3, 20a3b2, 36a3b3
7. Returning to practice problem 13, suppose the bakery packed its cookies in boxes of 20
cookies and boxes of 32 cookies. How many cookies should be paced in each cellophane
package?
8. Renovations Ms. Baxter wants to tile a wall to serve as a splashguard above a basin in the
basement. She plans to use equal-sized tiles to cover an area that measures 48 inches by 36
inches.
a. What is the maximum-size square tile Ms. Baxter can use and not have to cut any of the
tiles?
b. How many tiles of this size will she need?
9. Mathematicians A Greek mathematician and astronomer named Eratosthenes created a way
to separate prime numbers from composite numbers. His method is known as the Sieve of
Eratosthenes. It proceeds as follows.
a. Write the numbers 1 to 50.
b. Since 1 is neither prime nor composite, ignore 1.
c. Circle the number 2 and then cross off every number that is divisible by 2.
d. Circle the next number that is not crossed off (3) and cross off all multiples of 3. Circle the next
number that is not crossed off (5) and cross off the multiples of 5, etc…
Recreate the Sieve of Eratosthenes to find the first 11 prime numbers.
Section 2 – Factoring Using the Distributive Property
Essential Question: Many situations in real-life can be represented by quadratic equations. How
can these quadratic equations be solved using factoring to determine specific values in the given
situation?
Objective: Generate the factorization of polynomials using the greatest common factor and the
distributive property.
Bell-ringer:
Find the prime factorization of each number.
1. 86
2. 168
Find the Greatest Common Factor of each set of monomials.
3. 49 and 77
4. 24 and 104
5. 40xy2 and 64yz2
Factoring:
Practice - Use the distributive property to factor each polynomial.
1. 12mn2 – 18m2n2
2. 20abc + 15a2c – 5ac
3. 25a4 + 15a2
4. 28a2b + 56abc2
Solve each equation.
5. x(x – 32) = 0
7. 8p2 – 4p = 0
6. 4b(b + 4) = 0
8. 2z2 + 20z = 0
Where might we use this?
6. Swimming Pool The area A of a rectangular swimming pool is given by the equation A = 12w –
w2, where w is the width of one side. Write an expression for the other side of the pool.
7. Vertical Jump Your vertical jump height is measured by subtracting your standing reach
height from the height of the highest point you can reach by jumping without taking a running
start. Typically, NBA players have vertical jump heights of up to 34 inches. If an NBA player
jumps this high, his height h in inches above his standing reach height after t seconds can be
modeled by the equations h = 162t – 192t2. Solve the equation for h = 0 and interpret the
solution. Round you answer to the nearest hundredth.
Factoring by grouping:
Practice – Factor.
8. x2 + 4x + 2x + 8
9. 6x2y + 9xyz + 4xy2 + 6y2z
10. 49m + 21n2 + 35m3n + 15m2n3
11. 12ac + 21ad + 8bc + 14bd
12. 6y2 – 4y + 3y – 2
13. xa + ya + x + y
Ticket-Out: (To be completed in composition book) Factor each polynomial.
1. 8ax – 56a
2. a2b2 + a
3. x2 + 3x + x + 3
4. 9x2 – 3xy + 6x -2y
Independent Practice – Factor each polynomial.
1. 10a2 + 40a
2. 15wx – 35wx2
3. 16y2 + 8y
4. 2m3n2 – 16m2n3 + 8mn
5. 2ax + 6xc + ba + 3bc
6. 6mx – 4m + 3rx – 2r
7. 8ac – 2ad + 4bc – bd
8. 2e2g + 2fg + 4e2h + 4fh
9. Pets Conner tosses a dog treat upward with an initial velocity of 13.7 meters per second. The
height of the treat above the dog’s mouth h in meters after t seconds is given by the equation
h = 13.7t – 4.9t2.
a. Assuming the dog doesn’t jump, after how many seconds does the dog catch the treat?
b. The dog treat reaches its maximum height halfway between when it was thrown and
when it was caught. What is its maximum height?
c. How fast would Connor have to throw the dog treat in order to make it fly through the
air for 6 seconds?
Section 3 – Factoring Trinomials
Essential Question: Many situations in real-life can be represented by quadratic equations. How
can these quadratic equations be solved using factoring to determine specific values in the given
situation?
Objective: Generate the factorization of trinomials using the guess and check method.
Bell-ringer:
1. Factor 10xy + 25x – 14y -35.
2. In each diagram below, write the two numbers on the sides of the “X” that are multiplied
together to get the top number of the “X,” but added together to get the bottom number of
the “X.”
a.
b.
c.
9
4
-6
4
d.
e.
-30
-13
f.
-84
-24
6
5
-5
-5
Guess and Check:
Multiply (5x + 3)(2x + 7) using the FOIL method.
What is the product of the first and last coefficients? How does this compare to the product of the
middle two coefficients before the polynomial was simplified?
We will use this relationship to factor quadratic (degree two) polynomials.
Practice – Factor following the steps below.
1. Is there a GCF? If so, factor it out.
2. What is the product of the first and last coefficients?
3. What other numbers have the same product and the sum of the middle number?
4. Rewrite the middle term as the sum of those two numbers.
5. Distribute the variable to each new coefficient.
6. Factor by grouping.
1. 2x2 + 9x + 10
2. 3a2 + 13a + 4
3. 18t3 + 75t2 + 42t
4. 15y2 + 34y + 15
5. b2 + 7b + 12
6. 2s2 + 6s – 260
7. 4h2 + 8h – 5
8. 6q2 – 13q + 6
Some trinomials cannot be factored and are, therefore, considered prime polynomials.
Practice – Determine which of the following trinomials can be factored and factor it. If it cannot
be factored, write prime.
7. t2 -2t + 35
8. 2x2 – 5x – 12
9. 72 – 26y + 2y2
10. 3c2 -3c - 5
Where might we use this?
11. Compact Discs A standard jewel case for a compact disc has a width of 2 cm greater than its
length. The area for the front cover is 168 square centimeters. The first two steps to finding
the value of x are shown below. Solve the equation and find the length of the case.
Length x width = area
x( x + 2) = 168
x2 + 2x – 168 = 0
12. Bridge Engineering A car driving over a suspension bridge is supported by a cable hanging
between the ends of the bridge. Since its shape is parabolic, it can be modeled by a quadratic
equation. The height above the road bed of a bridge’s cable h
(in inches) measured at distance d (in yards) from the first
tower is given by the equation h = d2 – 36d + 324.
If the driver of a car looks out at a height of 49 inches above the
roadbed, at what distance(s) from the tower will the driver’s
eyes be at the same height as the cable?
Ticket-out: (To be completed in composition book.) Factor each polynomial.
Factor each polynomial.
1. x2 – 6x – 55
2. m2 – m – 56
3. 9p2 + 6p -8
4. 2t2 – 11t + 15
Independent Practice – Factor each trinomial if possible. If the trinomial cannot be factored
using integers, write prime.
1. 5x2 – 17x + 14
2. x2 + 2x – 15
3. b2 +22b + 21
4. 2n2 – 11n + 7
5. 8m2 – 10m + 3
6. 2r2 + 3r – 14
7. 5r2 – 3r + 15
8. a2 – 9a – 36
9. 4k2 + 2k – 12
10. Physical Science The boiling point of water depends on altitude. The following equation
approximates the number of degrees d below 212F at which water will boil at altitude h.
d2 + 520d = H
In Denver, Colorado, the altitude is approximately 5300 feet above sea level. At
approximately what temperature does water boil in Denver?
Section 4 – Factoring Differences of Squares
Essential Question: Many situations in real-life can be represented by quadratic equations. How
can these quadratic equations be solved using factoring to determine specific values in the given
situation?
Objective: Generate the factorization of difference of squares polynomials.
Bell-ringer:
1. Furniture The student council wants to purchase a table for the school lobby. The table
comes in a variety of dimensions, but for every table, the length is 1 meter greater than twice
the width. The student council has budgeted for a tabletop with an area of 3 square meters.
2w + 1
Find the width and length
2. Multiply each pair of binomials.
a. (x – 4)(x + 4)
b. (m + 10)(m – 10)
w
of the table they can purchase.
c. (w – 16)(w + 16)
What do we call the type of answers that you received for a, b, and c?
Difference of Squares:
Practice - Factor each binomial. Always check for a GCF first.
1. x2 – 25
2. a2 - 121
3. 16x4 – z4
4. 3k4 – 48
5. 81a2 – 16y2
6. a4 – 81b8
Pythagorean Triple:
Practice – Find a Pythagorean triple that includes 6 as one of its numbers.
Where might we use this?
7. Lottery A state lottery commission analyzes the ticket purchasing patterns of its citizens. The
following express is developed to help officials calculate the likely number of people who will
buy tickets for a certain size jackpot.
81a2 – 36b2
Factor the expression completely.
8. Optics A reflector on the inside of a certain flashlight is a parabola given by the equation
y = x2 – 25. Find the points where the reflector meets the lens by finding the values of x when
y = 0.
Ticket-out (To be completed in composition book.) Factor each polynomial.
1. a2 - 4
2. 4h2 – 25g2
3. -49r2 + 4t2
Independent Practice – Factor each polynomial if possible. If the polynomial cannot be factored
write prime. Remember to check for a GCF first.
1. x2 – 9
2. a2 – 64
3. t2 – 49
4. 4x2 – 9y2
5. 1 – 9z2
6. 16a2 – 9b2
7. 8x2 – 12y2
8. 75r2 – 48
9. 3a2 – 16
10. 12a2 – 48
11. -45m2 + 5
12. a2 + 4b2
13. Architecture The drawing shows a triangular roof truss with a base measuring
the same as its height. The area of the truss is 98 square meters. Find the height of
the truss.
height
base
14. Ballooning The function f(t) = -16t2 + 576 represents the height of a freely falling ballast bag
that starts from rest on a balloon 576 feet above the ground. After how many seconds t does the
ballast bag hit the ground?
Section 5 – Perfect Squares and Factoring
Essential Question: Many situations in real-life can be represented by quadratic equations. How
can these quadratic equations be solved using factoring to determine specific values in the given
situation?
Objective: Generate the factorization using perfect squares.
Bell-ringer:
1. Forensics Mr. Cooper contested a speeding ticket given to him after he applied his brakes and
skidded to a halt to avoid hitting another car. In traffic court, he argued that the length of the
skid marks on the pavement, 150 feet, proved that he was driving under the posted speed
limit of 65 miles per hour. The ticket cited his speed at 70 miles per hour. Use the formula s2
= 24d, where s is the speed of the car and d is the length of the skid marks, to determine Mr.
Cooper’s speed when he applied the brakes. Was Mr. Cooper correct in claiming that he was
not speeding when he applied the brakes?
2. Multiply each pair of binomials.
a. (x – 4)2
b. (m + 6)2
What do we call the problems that we solved in #2?
Perfect Square Trinomials:
Practice – Determine whether each trinomial is a perfect square trinomial. If so, factor it.
1. x2 – 12x + 36
2. a2 + 14a – 49
If one of the trinomials above was not a perfect square trinomial, how can it be changed to make
it one?
Determine all values of k that make each trinomial a perfect square trinomial.
3. 9x2 + kx + 16
4. 49x2 + kx + 9
Where might we use this?
5. Construction The area of Liberty Township’s square playground is represented by the
trinomial x2 – 10x + 25. Write an expression using the variable x that represents the perimeter.
Ticket-out (To be completed in composition book.) Factor each polynomial.
1. c2 – 6c + 9
2. r2 + 4r + 4
3. 25 + 30n + 9n2
Independent Practice – Determine whether each trinomial is a perfect square trinomial. If so,
factor it. Remember to check for a GCF first.
1. x2 + 12x + 36
2. n2 – 13n + 36
3. a2 + 4a + 4
4. 9b2 – 6b + 1
5. 4x2 + 4x + 1
6. 2n2 + 17n + 21
7. 2b2 – 28b + 98
8. 9y2 + 8y – 16
9. 5y2 – 50y + 500
Factoring Review – Use any method of factoring to factor each polynomial. As always, check for
GCF first.
10. n2 – 8n + 16
11. c2 – 15c + 56
12. 3g2 – 7g + 2
13. 32 – 8y2
14. 8x2 + 50
15. 10w2 – 19w – 15
16. Business Saini Sprinkler Company installs irrigation systems. To track monthly cost C and
revenues R, they use the following functions, where x is the number of systems they install.
R(x) = 8x2 + 12x + 4
C(x) = 7x2 + 20x – 12
The monthly profit can be found by subtracting cost from revenue.
P(x) = R(x) – C(x)
Find a function to project monthly profit and use it to find the break-even point where the
profit is zero
Section 6 – Solving Equations by Factoring
Essential Question: Many situations in real-life can be represented by quadratic equations. How
can these quadratic equations be solved using factoring to determine specific values in the given
situation?
Objective: Apply the concepts of factoring to solve equations, including equations representing
real-life situations.
Bell-ringer: Factor each polynomial.
1. 16x4 – 36x2y2
3. 6t3 -21t2 – 45t
2. 20a2 – 7a – 6
4. 4n3 – 20n2 – 9n + 45
5. The measure of the area of a rectangle is represented by 20a2 – 13a - 15. Find the measures of
its dimensions.
Throughout the chapter, we have been solving real-life problems using equations and factoring.
To do this we have applied the zero product property.
Zero product property:
Practice - Solve each equation.
1. (x + 3)(x – 5) = 0
2. (2a + 4)(a + 7) = 0
3. x2 – 36 = 5x
4. a2 – 24a = - 144
5. 4m2 + 25 = 20m
6. x3 + 2x2 = 15x
7. a3 – 13a2 + 42a = 0
8. 6t3 + t2 – 5t = 0
9. Football If h = vt – 16t2 and a punter can kick a football with an initial velocity of 48 feet per
second, how many seconds will it take for the ball to return to the ground?
Ticket-out: (To be completed in composition book). Solve each equation.
1. x2 – 18x + 81 = 0
2. 4y2 = 81
3. 5d2 – 22d + 8 = 0
Independent Practice – Solve each equation.
1. y(y – 12) = 0
2. 2x(5x – 10) = 0
3. (3x – 5)2 = 0
4. x2 – 6x = 0
5. 2x2 + 4x = 0
6. 2x2 = x2 – 8x
7. n2 – 3n = 0
8. n2 + 36n = 0
9. 8a2 = -4a
10. Geometry The length of a Charlotte, North Carolina conservatory garden is 20 yards greater
than its width. The area is 300 square yards. What are the dimensions?