Algebra 9.4

Warm-Up Exercises Lesson 9.4 Part 1

1. Find the GCF of 12 and 28.

2. Find the GCF of 18 and 42.

ANSWER 4

ANSWER 6

Warm-Up Exercises

ANSWER about 10,070

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?

3.

Lesson 9.4

Warm-Up ExercisesUse the zero-product propertyEXAMPLE 1

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.

Warm-Up ExercisesUse the zero-product propertyEXAMPLE 1

Solve (x – 4)(x + 2) = 0.

(x – 4)(x + 2) = 0 Write original equation.

x – 4 = 0 x = 4

Zero-product property

Solve for x.

ANSWER

The solutions of the equation are 4 and –2.

oror x + 2 = 0

x = – 2

Warm-Up ExercisesGUIDED PRACTICE for Example 1

1. Solve the equation (x – 5)(x – 1) = 0.

(x – 5)(x – 1) = 0 Write original equation.

x – 5 = 0

x = 5


Solve for x.

ANSWER

The solutions of the equation are 5 and 1.

or

or x – 1 = 0

x = 1

Warm-Up Exercises

SOLUTION

EXAMPLE 2 Find the greatest common monomial factor

Factor out the greatest common monomial factor.

a. 12x + 42y

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)

You may need to factor the polynomial before you can use theZero-Product Property to solve the equation. To factor it, look for a GCF (a monomial with an integer coefficient) that divides EVENLYinto each term.

Warm-Up ExercisesEXAMPLE 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)

SOLUTION

Factor out the greatest common monomial factor.

b. 4x4 + 24x3


2. Factor out the greatest common monomial factorfrom 14m + 35n.

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.

SOLUTION

ANSWER

14m + 35n = 7(2m + 5n)

Warm-Up ExercisesEXAMPLE 3 Solve an equation by factoring

Solve 2x2 + 8x = 0 by factoring out the GCF first.

2x2 + 8x = 0.

2x(x + 4) = 0

2x = 0

x = 0

or x + 4 = 0

or x = – 4

ANSWER

The solutions of the equation are 0 and – 4.

Solve for x.


Factor left side.

Write original equation.

Warm-Up ExercisesEXAMPLE 4 Solve an equation by factoring

Solve 6n2 = 15n. First there needs to be a zero on one side.

6n2 – 15n = 0

3n(2n – 5) = 0

3n = 0

n = 0

2n – 5 = 0

n =52

or

or Solve for n.


Factor left side.

Subtract 15n from each side.

ANSWER

The solutions of the equation are 0 and52 .

Warm-Up ExercisesGUIDED PRACTICE for Examples 3 and 4

Solve the equation by factoring out the GCF first.

a2 + 5a = 0

a(a + 5) = 0

a = 0

a = 0

or a + 5 = 0

or a = – 5

ANSWER

The solutions of the equation are 0 and – 5.

Solve for x.


Factor left side.


3. a2 + 5a = 0.


3s2 – 9s = 0

3s(s – 3) = 0

3s = 0

s= 0

or s – 3 = 0

or s = 3

ANSWER

The solutions of the equation are 0 and 3.

Solve for x.


Factor left side.


4. 3s2 – 9s = 0.


5. Solve 4x2 = 2x. Make sure there is a zero on one side first.

4x2 – 2x = 0

2x(2x – 1) = 0

2x = 0

x = 0

2x – 1 = 0

x =12

or

or Solve for x.


Factor left side.

Subtract 2x from each side.

ANSWER

The solutions of the equation are 0 and12 .

4x2 = 2x Write original equation.

Warm-Up Exercises

ARMADILLO

EXAMPLE 5 Solve a multi-step problem

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

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). €

2

Warm-Up Exercises

SOLUTION

EXAMPLE 5 Solve a multi-step problem

STEP 1

Write a model for the armadillo’s height above the ground.

h = – 16t2 + vt + s

h = – 16t2 + 14t + 0

h = – 16t2 + 14t

Vertical motion model

Substitute 14 for v and 0 for s.

Simplify.

Warm-Up ExercisesEXAMPLE 5 Solve a multi-step problem

STEP 2Substitute 0 for h. When the armadillo lands, its height above the ground is 0 feet. Solve for t.

0 = – 16t2 + 14t

0 = 2t(–8t + 7)

2t = 0

t = 0

–8t + 7 = 0

t = 0.875

or

or Solve for t.


Factor right side.

Substitute 0 for h.

ANSWER

The armadillo lands on the ground 0.875 second after the armadillo jumps.


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 1Write a model for the armadillo’s height above the ground.

h = – 16t2 + vt + s

h = – 16t2 + 12t + 0

Vertical motion model

Substitute 12 for v and 0 for s.

h = – 16t2 + 12t Simplify.


STEP 2Substitute 0 for h. When the armadillo lands, its height above the ground is 0 feet. Solve for t.

0 = – 16t2 + 12t

0 = – 4t(4t – 3)

– 4t = 0

t = 0

4t – 3 = 0

t = 0.75

or

or Solve for t.


Factor right side.

Substitute 0 for h.

ANSWER

The armadillo lands on the ground 0.75 second after the armadillo jumps.

Warm-Up ExercisesLesson Review Part 1

1. (y + 5 ) (y – 9 ) = 0

ANSWER – 5 , 9

2. (2n + 3 ) (n – 4 ) = 0

ANSWER 32

– , 4

3. 6x2 =20x

ANSWER103

0,

Solve the equation by finding the roots.

Warm-Up ExercisesLesson Review For use after Lesson 9.4

4. 12x2 =18x

ANSWER32

0,

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

Warm-Up Exercises Homework

HomeworkDue Thursday 3/8

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Algebra 9.4