The document provides examples and practice problems for solving polynomial equations by factoring polynomials to find their roots using the zero-product property. Students are given examples of factoring polynomials to find their greatest common factor before setting each factor equal to zero to solve for the roots of the original equation. Practice problems include multi-step word problems involving vertical motion to solve for time using a quadratic model.

1. Warm-Up Exercises Lesson 9.4 Part 1
1. Find the GCF of 12 and 28.
ANSWER 4
2. Find the GCF of 18 and 42.
ANSWER 6

2. Warm-Up Exercises Lesson 9.4
3. The number (in hundreds) of sunscreen and sun
tanning products sold at a pharmacy from 2005-2011
can be modeled by –0.8t2 + 0.3t + 107, where t is the
number of years since 2005.
About how many products were sold in 2008?
ANSWER about 10,070

3. EXAMPLE 1Exercises zero-product property
Warm-Up Use the
Need to know!
*The solutions of a Polynomial Equation are called roots.
*A Polynomial Equation is an equation where one side of the equal
sign is a product of polynomial factors and the other side is 0.
Example: (x + 2)(x - 6) = 0
The Zero-Product Property is used to solve polynomial equations.
It states that one of the polynomials must be equal to zero if the
whole equation is equal to zero.

4. EXAMPLE 1Exercises zero-product property
Warm-Up Use the
Solve (x – 4)(x + 2) = 0.
(x – 4)(x + 2) = 0 Write original equation.
x – 4 = 0 or x + 2 = 0 Zero-product property
x = 4 or x=–2 Solve for x.
ANSWER
The solutions of the equation are 4 and –2.

5. Warm-Up Exercises
GUIDED PRACTICE for Example 1
1. Solve the equation (x – 5)(x – 1) = 0.
(x – 5)(x – 1) = 0 Write original equation.
x – 5 = 0 or x – 1 = 0 Zero-product property
x = 5 or x=1 Solve for x.
ANSWER
The solutions of the equation are 5 and 1.

6. EXAMPLE 2Exercises greatest common monomial factor
Warm-Up Find the
You may need to factor the polynomial before you can use the
Zero-Product Property to solve the equation. To factor it, look
for a GCF (a monomial with an integer coefficient) that divides EVENLY
into each term.
Factor out the greatest common monomial factor.
a. 12x + 42y
SOLUTION
a. The GCF of 12 and 42 is 6. The variables x and y
have no common factor. So, the greatest
common monomial factor of the terms is 6.
ANSWER
12x + 42y = 6(2x + 7y)

7. EXAMPLE 2Exercises greatest common monomial factor
Warm-Up Find the
Factor out the greatest common monomial factor.
b. 4x4 + 24x3
SOLUTION
b. The GCF of 4 and 24 is 4. The GCF of x4 and x3 is x3.
So, the greatest common monomial factor of the
terms is 4x3.
ANSWER
4x4 + 24x3 = 4x3(x + 6)

8. Warm-Up Exercises
GUIDED PRACTICE for Example 2
2. Factor out the greatest common monomial factor
from 14m + 35n.
SOLUTION
The GCF of 14 and 35 is 7. The variables m and n have
no common factor. So, the greatest common
monomial factor of the terms is 7.
ANSWER
14m + 35n = 7(2m + 5n)

9. EXAMPLE 3Exercises equation by factoring
Warm-Up Solve an
Solve 2x2 + 8x = 0 by factoring out the GCF first.
2x2 + 8x = 0. Write original equation.
2x(x + 4) = 0 Factor left side.
2x = 0 or x + 4 = 0 Zero-product property
x=0 or x=–4 Solve for x.
ANSWER
The solutions of the equation are 0 and – 4.

10. EXAMPLE 4Exercises equation by factoring
Warm-Up Solve an
Solve 6n2 = 15n. First there needs to be a zero on one side.
6n2 – 15n = 0 Subtract 15n from each side.
3n(2n – 5) = 0 Factor left side.
3n = 0 or 2n – 5 = 0 Zero-product property
5
n=0 or n= Solve for n.
2
ANSWER
5
The solutions of the equation are 0 and .
2

11. Warm-Up Exercises
GUIDED PRACTICE for Examples 3 and 4
Solve the equation by factoring out the GCF first.
3. a2 + 5a = 0.
a2 + 5a = 0 Write original equation.
a(a + 5) = 0 Factor left side.
a=0 or a + 5 = 0 Zero-product property
a=0 or a=–5 Solve for x.
ANSWER
The solutions of the equation are 0 and – 5.

12. Warm-Up Exercises
GUIDED PRACTICE for Examples 3 and 4
4. 3s2 – 9s = 0.
3s2 – 9s = 0 Write original equation.
3s(s – 3) = 0 Factor left side.
3s = 0 or s – 3 = 0 Zero-product property
s= 0 or s=3 Solve for x.
ANSWER
The solutions of the equation are 0 and 3.

13. Warm-Up Exercises
GUIDED PRACTICE for Examples 3 and 4
5. Solve 4x2 = 2x. Make sure there is a zero on one side first.
4x2 = 2x Write original equation.
4x2 – 2x = 0 Subtract 2x from each side.
2x(2x – 1) = 0 Factor left side.
2x = 0 or 2x – 1 = 0 Zero-product property
1
x=0 or x= Solve for x.
2
ANSWER
1
The solutions of the equation are 0 and .
2

14. EXAMPLE 5Exercises multi-step problem
Warm-Up Solve a
ARMADILLO
A startled armadillo jumps
straight into the air with an
initial vertical velocity of 14
feet per second.
After how many seconds
does it land on the ground?
Vertical Motion Formula
2
h = -16t + vt + s
where t is the time (sec.) the object has
been in the air, v is the initial vertical
velocity (ft./sec.), and s is the initial
height (feet).

15. EXAMPLE 5Exercises multi-step problem
Warm-Up Solve a
SOLUTION
STEP 1
Write a model for the armadillo’s height above the
ground.
h = – 16t2 + vt + s Vertical motion model
h = – 16t2 + 14t + 0 Substitute 14 for v and 0 for s.
h = – 16t2 + 14t Simplify.

16. EXAMPLE 5Exercises multi-step problem
Warm-Up Solve a
STEP 2
Substitute 0 for h. When the armadillo lands, its height
above the ground is 0 feet. Solve for t.
0 = – 16t2 + 14t Substitute 0 for h.
0 = 2t(–8t + 7) Factor right side.
2t = 0 or –8t + 7 = 0 Zero-product property
t=0 or t = 0.875 Solve for t.
ANSWER
The armadillo lands on the ground 0.875 second after
the armadillo jumps.

17. Warm-Up Exercises
GUIDED PRACTICE for Example 5
6. WHAT IF? In Example 5, suppose the initial
vertical velocity is 12 feet per second.
After how many seconds does armadillo land on the
ground?
SOLUTION
STEP 1
Write a model for the armadillo’s height above the
ground.
h = – 16t2 + vt + s Vertical motion model
h = – 16t2 + 12t + 0 Substitute 12 for v and 0 for s.
h = – 16t2 + 12t Simplify.

18. Warm-Up Exercises
GUIDED PRACTICE for Example 5
STEP 2
Substitute 0 for h. When the armadillo lands, its height
above the ground is 0 feet. Solve for t.
0 = – 16t2 + 12t Substitute 0 for h.
0 = – 4t(4t – 3) Factor right side.
– 4t = 0 or 4t – 3 = 0 Zero-product property
t=0 or t = 0.75 Solve for t.
ANSWER
The armadillo lands on the ground 0.75 second after
the armadillo jumps.

19. Lesson Review
Warm-Up Exercises Part 1
Solve the equation by finding the roots.
1. (y + 5 ) (y – 9 ) = 0
ANSWER –5,9
2. (2n + 3 ) (n – 4 ) = 0
ANSWER 3 ,4
–
2
3. 6x2 =20x
ANSWER 0, 10
3

20. Lesson Review
Warm-Up Exercises For use after Lesson 9.4
4. 12x2 =18x
ANSWER 0, 3
2
5. A dog jumps in the air with an initial velocity of
18 feet per second to catch a flying disc. How
long does the dog remain in the air?
Use h = – 16t2 + vt + s
ANSWER 1.125 sec

21. Homework
Warm-Up Exercises
Homework
Due Thursday 3/8
Pages 578 - 579
54, 45 - 3 (x3)
REVIEW TOMORROW & QUIZ FRIDAY
Sections 1 - 4