Text 9. Factoring

ALGEBRA PROJECT UNIT 9 FACTORING

FACTORING

Lesson 1

Factors and Greatest Common Factors

Lesson 2

Factoring Using the Distributive Property

Lesson 3

Factoring Trinomials: x2 + bx + c

Lesson 4

Factoring Trinomials: ax2 + bx + c

Lesson 5

Factoring Differences of Squares

Lesson 6

Perfect Squares and Factoring

FACTORS And GREATEST COMMON FACTORS

Example 1

Classify Numbers as Prime or Composite

Example 2

Prime Factorization of a Positive Integer

Example 3

Prime Factorization of a Negative Integer

Example 4

Prime Factorization of a Monomial

Example 5

GCF of a Set of Monomials

Example 6

Use Factors

Factor 22. Then classify it as prime or composite. To find the factors of 22, list all pairs of whole numbers whose product is 22.

Answer: Since 22 has more than two factors, it is a composite number. The factors of 22, in increasing order, are 1, 2, 11, and 22.

Factor 31. Then classify it as prime or composite. The only whole numbers that can be multiplied together to get 31 are 1 and 31. Answer: The factors of 31 are 1 and 31. Since the only factors of 31 are 1 and itself, 31 is a prime number.

Factor each number. Then classify it as prime or composite. a. 17 Answer: 1, 17; prime

b. 25 Answer: 1, 5, 25; composite

Find the prime factorization of 84. Method 1 The least prime factor of 84 is 2. The least prime factor of 42 is 2. The least prime factor of 21 is 3. All of the factors in the last row are prime. Answer: Thus, the prime factorization of 84 is

Method 2 Use a factor tree.

84 21 3

4 7

2

2

and

All of the factors in the last branch of the factor tree are prime. Answer: Thus, the prime factorization of 84 is or

Find the prime factorization of 60.

Answer:

or

Find the prime factorization of –132. Express –132 as –1 times 132.

/ \ / \ / \

Answer: The prime factorization of –132 is or

Find the prime factorization of –154. Answer:

Factor

Answer:

completely.

in factored form is

Factor

completely. Express –26 as –1 times 26.

Answer:

in factored form is

Factor each monomial completely. a. Answer:

b. Answer:

Find the GCF of 12 and 18. Factor each number. Circle the common prime factors. The integers 12 and 18 have one 2 and one 3 as common prime factors. The product of these common prime factors, or 6, is the GCF. Answer: The GCF of 12 and 18 is 6.

.

Find the GCF of

Factor each number. Circle the common prime factors. Answer: The GCF of

and

is

.

Find the GCF of each set of monomials. a. 15 and 35 Answer: 5 b. Answer:

and

Crafts Rene has crocheted 32 squares for an afghan. Each square is 1 foot square. She is not sure how she will arrange the squares but does know it will be rectangular and have a ribbon trim. What is the maximum amount of ribbon she might need to finish an afghan? Find the factors of 32 and draw rectangles with each length and width. Then find each perimeter. The factors of 32 are 1, 2, 4, 8, 16, 32.

The greatest perimeter is 66 feet. The afghan with this perimeter has a length of 32 feet and a width of 1 foot. Answer: The maximum amount of ribbon Rene will need is 66 feet.

Mary wants to plant a rectangular flower bed in her front yard with a stone border. The area of the flower bed will be 45 square feet and the stones are one foot square each. What is the maximum number of stones that Mary will need to go around all four sides of the flower bed?

Answer: 92 feet

FACTORING USING THE DISTRIBUTIVE PROPERTY

Example 1

Use the Distributive Property

Example 2

Use Grouping

Example 3

Use the Additive Inverse Property

Example 4

Solve an Equation in Factored Form

Example 5

Solve an Equation by Factoring

Use the Distributive Property to factor First, find the CGF of 15x and . Factor each number.

.

Circle the common prime factors. GFC: Write each term as the product of the GCF and its remaining factors. Then use the Distributive Property to factor out the GCF. Rewrite each term using the GCF. Simplify remaining factors. Distributive Property

Answer: The completely factored form of is

Use the Distributive Property to factor . Factor each number. Circle the common prime factors. GFC:

or Rewrite each term using the GCF.

Distributive Property Answer: The factored form of is

Use the Distributive Property to factor each polynomial. a. Answer:

b. Answer:

Factor Group terms with common factors. Factor the GCF from each grouping. Answer:

Distributive Property

Factor Answer:

Factor Group terms with common factors. Factor GCF from each grouping.

Answer: Distributive Property

Factor Answer:

Solve

Then check the solutions.

If Property either

, then according to the Zero Product or Original equation

or

Set each factor equal to zero. Solve each equation.

Answer: The solution set is

Check Substitute 2 and

for x in the original equation.

Solve Answer: {3, –2}

Then check the solutions.

Solve

Then check the solutions.

Write the equation so that it is of the form Original equation Subtract

from each side.

Factor the GCF of 4y and which is 4y. or

Zero Product Property Solve each equation.

Answer: The solution set is

0 and

Check by substituting

for y in the original equation.

Solve Answer:

FACTORING TRINOMIALS X + bX + c 2

Example 1

b and c Are Positive

Example 2

b Is Negative and c Is Positive

Example 3

b Is Positive and c Is Negative

Example 4

b Is Negative and c Is Negative

Example 5

Solve an Equation by Factoring

Example 6

Solve a Real-World Problem by Factoring

Factor In this trinomial, and You need to find the two numbers whose sum is 7 and whose product is 12. Make an organized list of the factors of 12, and look for the pair of factors whose sum is 7. Factors of 12

Sum of Factors

1, 12 2, 6 3, 4

13 8 7

Answer:

The correct factors are 3 and 4. Write the pattern. and

Check You can check the result by multiplying the two factors. F

O

I

L FOIL method Simplify.

Factor Answer:

Factor In this trinomial, and This means is negative and mn is positive. So m and n must both be negative. Therefore, make a list of the negative factors of 27, and look for the pair whose sum is –12. Factors of 27

Sum of Factors

–1, –27 –3, –9

–28 –12

Answer:

The correct factors are –3 and –9. Write the pattern. and

Check You can check this result by using a graphing calculator. Graph and on the same screen. Since only one graph appears, the two graphs must coincide. Therefore, the trinomial has been factored correctly.

Factor Answer:

Factor In this trinomial, and This means is positive and mn is negative, so either m or n is negative, but not both. Therefore, make a list of the factors of –18 where one factor of each pair is negative. Look for the pair of factors whose sum is 3. Factors of –18

Sum of Factors

1, –18 –1, 18 2, –9 –2, 9 3, –6 –3, 6

–17 17 – 7 7 – 3 3

The correct factors are –3 and 6.

Write the pattern. Answer:

and

Factor Answer:

Factor Since and is negative and mn is negative. So either m or n is negative, but not both. Factors of –20

Sum of Factors

1, –20 –1, 20 2, –10 –2, 10 4, –5 –4, 5

–19 19 – 8 8 – 1 1

The correct factors are 4 and –5.

Answer:

Write the pattern. and

Factor Answer:

Solve

Check your solutions. Original equation Rewrite the equation so that one side equals 0. Factor. or

Zero Product Property Solve each equation.

Answer: The solution is

Check Substitute –5 and 3 for x in the original equation.

Solve Answer:

Check your solutions.

Architecture Marion has a small art studio measuring 10 feet by 12 feet in her backyard. She wants to build a new studio that has three times the area of the old studio by increasing the length and width by the same amount. What will be the dimensions of the new studio? Explore Begin by making a diagram like the one shown to the right, labeling the appropriate dimensions.

Plan

Let the amount added to each dimension of the studio. The new length times the new width equals the new area.

old area

Solve

Write the equation. Multiply. Subtract 360 from each side.

Factor. or

Zero Product Property Solve each equation.

Examine

The solution set is Only 8 is a valid solution, since dimensions cannot be negative.

Answer:

The length of the new studio should be or 20 feet and the new width should be or 18 feet.

Photography Adina has a photograph. She wants to enlarge the photograph by increasing the length and width by the same amount. What dimensions of the enlarged photograph will be twice the area of the original photograph? Answer:

FACTORING TRINOMIALS aX + bX + c 2

Example 1

Factor ax2 + bx + c

Example 2

Factor When a, b, and c Have a Common Factor

Example 3

Determine Whether a Polynomial Is Prime

Example 4

Solve Equations by Factoring

Example 5

Solve Real-World Problems by Factoring

Factor In this trinomial, and You need to find two numbers whose sum is 27 and whose product is or 50. Make an organized list of factors of 50 and look for the pair of factors whose sum is 27. Factors of 50

Sum of Factors

1, 50 2, 25

51 27

The correct factors are 2 and 25.

Write the pattern. and Group terms with common factors. Factor the GCF from each grouping. Distributive Property

Answer:

Check You can check this result by multiplying the two factors. F

O

I

L FOIL method Simplify.

Factor

Answer:

Factor In this trinomial, and Since b is negative, is negative. Since c is positive, mn is positive. So m and n must both be negative. Therefore, make a list of the negative factors of or 72, and look for the pair of factors whose sum is –22. Factors of 72

Sum of Factors

–1, –72 –2, –36 –4, –24 –4, –18

–73 –38 –27 –22

The correct factors are –4, –18.

Write the pattern. and Group terms with common factors. Factor the GCF from each grouping. Answer:

Distributive Property

a. Factor Answer:

b. Factor Answer:

Factor Notice that the GCF of the terms , and 32 is 4. When the GCF of the terms of a trinomial is an integer other than 1, you should first factor out this GCF. Distributive Property Now factor Since the lead coefficient is 1, find the two factors of 8 whose sum is 6. Factors of 8

Sum of Factors

1, 8 2, 4

9 6

The correct factors are 2 and 4.

Answer: So, complete factorization of

Thus, the is

Factor Answer:

Factor In this trinomial, and Since b is positive, is positive. Since c is negative, mn is negative, so either m or n is negative, but not both. Therefore, make a list of all the factors of 3(–5) or –15, where one factor in each pair is negative. Look for the pair of factors whose sum is 7. Factors of –15

Sum of Factors

–1, 15 1, –15 –3, 5 3, –5

14 –14 2 –2

There are no factors whose sum is 7. Therefore, cannot be factored using integers. Answer:

is a prime polynomial.

Factor Answer: prime

Solve Original equation Rewrite so one side equals 0. Factor the left side. or

Zero Product Property Solve each equation.

Answer: The solution set is

Solve Answer:

Model Rockets Ms. Nguyen’s science class built an airlaunched model rocket for a competition. When they testlaunched their rocket outside the classroom, the rocket landed in a nearby tree. If the launch pad was 2 feet above the ground, the initial velocity of the rocket was 64 feet per second, and the rocket landed 30 feet above the ground, how long was the rocket in flight? Use the equation

Vertical motion model Subtract 30 from each side. Factor out –4. Divide each side by –4. Factor or

Zero Product Property Solve each equation.

The solutions are

and

seconds. The first time

represents how long it takes the rocket to reach a height of 30 feet on its way up. The second time represents how long it will take for the rocket to reach the height of 30 feet again on its way down. Thus the rocket will be in flight for 3.5 seconds before coming down again. Answer: 3.5 seconds

When Mario jumps over a hurdle, his feet leave the ground traveling at an initial upward velocity of 12 feet per second. Find the time t in seconds it takes for Mario’s feet to reach the ground again. Use the equation

Answer:

second

FACTORING DIFFERENCE OF SQUARES

Example 1

Factor the Difference of Squares

Example 2

Factor Out a Common Factor

Example 3

Apply a Factoring Technique More Than Once

Example 4

Apply Several Different Factoring Techniques

Example 5

Solve Equations by Factoring

Example 6

Use Differences of Two Squares

Factor

. Write in form

Answer:

Factor the difference of squares.

Factor

. and

Answer:

Factor the difference of squares.

Factor each binomial. a. Answer:

b. Answer:

Factor The GCF of

and 27b is 3b.

and Answer:

Factor the difference of squares.

Factor Answer:

Factor The GCF of and 2500 is 4. and Factor the difference of squares. and Answer:

Factor the difference of squares.

Factor Answer:

Factor Original Polynomial Factor out the GCF. Group terms with common factors. Factor each grouping. is the common factor. Answer:

Factor the difference of squares, into .

Factor Answer:

Solve

by factoring. Check your solutions. Original equation. and Factor the difference of squares. or

Zero Product Property Solve each equation.

Answer: The solution set is Check each solution in the original equation.

Solve

by factoring. Check your solutions. Original equation Subtract 3y from each side. The GCF of

and 3y is 3y. and

Applying the Zero Product Property, set each factor equal to zero and solve the resulting three equations. or

or

Answer: The solution set is Check each solution in the original equation.

Solve each equation by factoring. Check your solutions. a. Answer:

b. Answer:

Extended-Response Test Item A square with side length x is cut from a right triangle shown below. a. Write an equation in terms of x that represents the area A of the figure after the corner is removed. b. What value of x will result in a figure that is

the area of the original

triangle? Show how you arrived at your answer.

Read the Test Item A is the area of the triangle minus the area of the square that is to be removed. Solve the Test Item a. The area of the triangle is the area of the square is

or 64 square units and square units.

Answer: b. Find x so that A is

the area of the original triangle, Translate the verbal statement.

and Simplify. Subtract 48 from each side. Simplify. Factor the difference of squares. or

Zero Product Property Solve each equation.

Answer: Since length cannot be negative, the only reasonable solution is 4.

Extended-Response Test Item A square with side length x is cut from the larger square shown below. a. Write an equation in terms of x that represents the area A of the figure after the corner is removed. Answer: b. What value of x will result in a figure that is

of the area of the

original square? Answer: 3

PERFECT SQUARES and FACTORING

Example 1

Factor Perfect Square Trinomials

Example 2

Factor Completely

Example 3

Solve Equations with Repeated Factors

Example 4

Use the Square Root Property to Solve Equations

Determine whether trinomial. If so, factor it. 1. Is the first term a perfect square? 2. Is the last term a perfect square? 3. Is the middle term equal to Answer:

is a perfect square Yes, Yes, ? Yes,

is a perfect square trinomial. Write as Factor using the pattern.

Determine whether square trinomial. If so, factor it. 1. Is the first term a perfect square? 2. Is the last term a perfect square? 3. Is the middle term equal to Answer:

is a perfect Yes, Yes, ? No,

is not a perfect square trinomial.

Determine whether each trinomial is a perfect square trinomial. If so, factor it. a. Answer: not a perfect square trinomial

b. Answer: yes;

Factor

.

First check for a GCF. Then, since the polynomial has two terms, check for the difference of squares.

6 is the GCF. and Answer:

Factor the difference of squares.

Factor

.

This polynomial has three terms that have a GCF of 1. While the first term is a perfect square, the last term is not. Therefore, this is not a perfect square trinomial. This trinomial is in the form Are there two numbers m and n whose product is and whose sum is 8? Yes, the product of 20 and –12 is –240 and their sum is 8.

Write the pattern. and Group terms with common factors. Factor out the GCF from each grouping.

Answer:

is the common factor.

Factor each polynomial. a. Answer:

b. Answer:

Solve Original equation Recognize as a perfect square trinomial. Factor the perfect square trinomial. Set the repeated factor equal to zero. Solve for x. Answer: Thus, the solution set is solution in the original equation.

Check this

Solve

Answer:

Solve

. Original equation Square Root Property Add 7 to each side. or

Separate into two equations. Simplify.

Answer: The solution set is solution in the original equation.

Check each

Solve

. Original equation Recognize perfect square trinomial. Factor perfect square trinomial. Square Root Property Subtract 6 from each side.

or

Separate into two equations. Simplify.

Answer: The solution set is solution in the original equation.

Check this

Solve

. Original equation Square Root Property Subtract 9 from each side.

Answer: Since 8 is not a perfect square, the solution set is Using a calculator, the approximate solutions are

or about –6.17 and

or about –11.83.

Check You can check your answer using a graphing calculator. Graph and Using the INTERSECT feature of your graphing calculator, find where The check of –6.17 as one of the approximate solutions is shown.

Solve each equation. Check your solutions. a. Answer: b Answer: c. Answer:

THIS IS THE END OF THE SESSION

BYE!