The document provides examples and explanations of the zero-product property, factoring polynomials, finding roots of polynomials, and solving equations involving vertical motion. It demonstrates using the zero-product property to solve equations by setting each factor equal to 0. It explains how to factor polynomials by finding the greatest common monomial factor. And it shows how to model and solve word problems involving the vertical motion of projectiles.

2. Zero-Product Property
 If ab 0 , then a
0 or b 0 .

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

4. Example 1
Use the zero-product property
CHECK Substitute each solution into the original
equation to check.
?
(4 – 4) (4 + 2) = 0
?
(– 2 – 4 ) (– 2 + 2 ) = 0
?
0 •6 = 0
?
–6 • 0 = 0
0 = 0
0 = 0

5. Factoring
 To solve a polynomial equation using the zero-product
property, factor the polynomial, which involves writing
it as a product of other polynomials.
 First step in factoring is to look for the greatest
common monomial factor of the polynomial’s terms.

6. Example 2
Find the greatest common monomial factor
Factor out the greatest common monomial factor.
a. 12x + 42y
b. 4x4 + 24x3
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 2
Find the greatest common monomial factor
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. Example 3
Solve an equation by factoring
Solve 2x2 = – 8x.
2x2 = – 8x
2x2 + 8x = 0
2x ( x + 4 ) = 0
2x = 0
or
x= 0
or
x +4 = 0
x = –4
Write original equation.
Add 8x to each side.
Factor left side.
Zero-product property
Solve for x.
ANSWER
The solutions of the equation are 0 and – 4.

9. Roots
 A root of polynomial involving x is a value of x for
which the corresponding value of the polynomial is 0.

10. Example 4
Find the roots of a polynomial
Find the roots of 6x2 – 15x.
6x2 – 15x = 0
Set polynomial equal to 0.
3x ( 2x – 5 ) = 0
Factor the polynomial.
3x = 0
or 2x – 5 = 0
x= 0
5
x =
2
or
Zero-product property
Solve for x.
ANSWER
5
The roots of the polynomial are 0 and .
2

11. Vertical Motion
 A projectile is an object that is propelled into the air
but has no power to keep itself in the air.
 A thrown ball is a projectile, but an airplane is not.
 The height of a projectile can be described by the
vertical motion model:
 The height h (in feet) of a projectile is modeled by
h
16t 2 vt s
 t is the time (in seconds) the object has been in the air.
 v is the initial vertical velocity (in feet per second).
 s is the initial height (in feet).

12. Example 5
Solve a multi-step problem
SALMON
As a salmon swims upstream, it leaps into the air with
an initial vertical velocity of 10 feet per second. After
how many seconds does the salmon return to the
water?
SOLUTION
STEP 1 Write a model for the salmon’s height above
the surface of the water.
h = – 16t2 + vt + s
Vertical motion model
h = – 16t2 + 10t + 0
Substitute 10 for v and 0 for s.
h = – 16t2 + 10t
Simplify.

13. Example 5
Solve a multi-step problem
STEP 2 Substitute 0 for h. When the salmon returns to
the water, its height above the surface is 0
feet. Solve for t.
0 = – 16t2 + 10t
Substitute 0 for h.
0 = 2t (– 8t + 5 )
Factor right side.
2t = 0 or – 8t + 5 = 0
t = 0 or
t = 0.625
Zero-product property
Solve for t.
ANSWER
The salmon returns to the water 0.625 second after
leaping.

14. 9.5 Warm-Up (Day 1)
Solve the equation
1. ( y 9)( y 1) 0
Factor out the greatest common monomial factor.
2. 3s 4 16s
Solve the equation
3. 10n 2 35n 0

15. 9.5 Warm-Up (Day 2)
Find the roots of the polynomial.
1. 6h2 3h
Factor out the greatest common monomial factor.
2. 20 x 2 y 2 4 xy
Solve the equation
3. 12 p 2
30 p