Chapter 1 — Real Numbers & Expressionslsimcik/math154/book/ch7.pdf · Chapter 7 — Factoring...

Chapter 7 — Factoring Caspers

Chapter 7 — Factoring

Section 7.1 — Greatest Common Factor

Section 7.2 — Factoring by Grouping

Section 7.3 — Factoring Trinomials with a Leading Coefficient of 1

Section 7.4 — Factoring a Difference of Squares

Section 7.5 — Factoring Trinomials with a non-1 Leading Coefficient

Section 7.6 — General Factoring

Section 7.7 — Solving Quadratic Equations by Factoring

Section 7.8 — General Application Problems Involving Quadratic Equations

Answers

Math 154 ::

Elementary Algebra

http://cabrillo.edu/~mcaspers/math154/book/answers/ch7.pdf

http://cabrillo.edu/~mcaspers/math154/book/answers/ch7.pdf

Math 154 :: Elementary Algebra Chapter 7 — Factoring

Section 7.1 — Greatest Common Factor 1 Caspers

Section 7.1 Greatest Common Factor

Examples:

Factor. Make sure each expression is factored completely. If the expression is not factorable, state that it is prime.

a) 4 2 2 3 440 8 16x y x y xy

To factor an expression means to write it as a product (multiplication). The first step is to determine if all terms have a

Greatest Common Factor (GCF). If there is a GCF, factor it out by “un-distributing it”. When determining the variable part of

a GCF, use the smallest exponent on that variable.

4 2 2 3 440 8 16x y x y xy The GCF for this expression is 28xy since all terms contain the factors of 8, x, and 2y .

2 3 28 5 2xy x xy y

b) 8 2 3 2 3x y y

Again, the first step when factoring is to determine if there is a Greatest Common Factor(GCF).

This expression has two terms: 8 2 3x y and 2 3y . The GCF of these two terms is 2 3y . To factor this expression,

it may be helpful to rewrite it: 8 2 3 1 2 3x y y

in order to “see” what factor remains in the second term when factoring out the GCF.

8 2 3 1 2 3x y y

2 3 8 1y x

Homework

1. In general, what does it mean to factor an expression?

2. In your own words, describe how to determine the greatest common factor of the terms in an expression.

3. If an expression has no greatest common factor, is it factorable?

4. What is an expression that isn’t factorable called?

5. In your own words, describe how to determine how many factors (the exponent) of any variables are in a greatest common factor.


6. 4 12x

7. 10 15p

8. 6 3y

9. 18 6x

10. 21 14k

11. 36 60y

12. 88 44n

13. 36 49x

14. 27 54x

15. 32 4y y

16. 4 316 8p p

17. 225 45x x


Section 7.1 — Greatest Common Factor 2 Caspers

18. 3 215 117a a

19. 6 430 9y y

20. 5 3 233 11 66x x x

21. 4 3 235 28 7p p p

22. 3 260 90x y xy

23. 3 2 380 72y z yz

24. 6 2 4 3 3 470 56 91a c a c a c

25. 5 3 2 2 465 13 39p p q p q

26. 4 3 248 64 8m n m n mn

27. 4 2 3 3 2 49 22 18x y x y x y

28. 4 2 2 352 26 39w z w z w

29. 3 2 2 345 60 90x y x y xy

30. 4 3 3 448 24 12a c a c ac

31. 1

23 6x x Hint: The GCF is

123x .

32. 1 2

3 314 2a a

33. 2 34 8x x Hint: The GCF is

34x .

34. 5 425 5y y

35. 5 4 5x a a

36. 1 7 1y z z

37. 7 3 2 3p q q

38. 3 4 4d c c

39. 9 2 9x x x

40. 3 3m m m


Section 7.2 — Factoring by Grouping 3 Caspers

2 2 2

GCF is 3 GCF is 4

3 6 4 8

y

x y y x y

2 23 2 4 2y x y x y

Section 7.2 Factoring by Grouping

7.2 — GCF and Grouping Worksheet

Example:


a) 2 2 23 6 4 8x y y x y

To factor an expression, first check for a GCF of all of the terms.

For this expression, there is not a GCF for all terms.

Next, count the number of terms in the expression. If there are four terms, try factoring by grouping. There are other

techniques for factoring expression with four terms, but this course does not cover those other methods.

To factor this expression by grouping, find the GCF of the first two terms and the GCF of the

second two terms.

Next, factor out each of these GCFs from the appropriate terms. Notice that for the second two

terms:

24 8x y , since the third term was negative, 4 was factored out. This leaves the

expression with two terms that have a GCF of 2 2x y .

2 2 3 4x y y Now, factor out the GCF of 2 2x y .This is the final answer. Notice that it is a product.

Homework


2. What are the two types of factoring methods covered in Chapter 7 so far (from sections 7.1 & 7.2)?

3. When asked to factor an expression, do you determine when to try factoring by grouping?

4. In your own words, describe how to factor an expression by grouping.

5. What factoring method should you always look for first, even if you eventually use another factoring method?

6. In your own words, describe how to determine if an expression has been factored completely.

7. What do you call an expression that cannot be factored using any factoring method?


8. 3 26 4 15 10x x x

9. 4 312 15 8 10p p p

10. 3 25 15 7 21y y y

11. 4 2 26 24 4k k k

12. 3 22 6 7 28n n n

13. 3 216 6 56 21x x x

14. 2 2ac c a

15. 3 218 72 10 40w w w Hint: GCF first! Watch for this in all problems.

16. 3 236 63 8 14y y y

17. 3 222 6 11 3x x x

http://cabrillo.edu/~mcaspers/math154/worksheets/7_2.pdf



Section 7.2 — Factoring by Grouping 4 Caspers

18. 230 80 9 24p p p

19. 3 29 81 9m m m

20. 4 2 27 7 7x x x

21. 2 23 6 4 8p q pq pq

22. 3 226 2 13 1m m m

23. 26 60 3 30x y xy xy y

24. 3 2 23 24 21 168a a a a

25. 2 2 2 22 3x y x y x x

26. 3 28 2 12 3p p p

27. 3 28 2 12 3y y y

28. 5 4 3 23 3x x x x


Section 7.3 — Factoring Trinomials with a Leading Coefficient of 1 5 Caspers

x x

8 3x x

Section 7.3 Factoring Trinomials with a Leading Coefficient of 1

Example:


a) 2 5 24x x

To factor an expression, the first check for a GCF of all of the terms.


There are three terms in the expression, so try factoring by “un-FOIL-ing”: find two binomials that multiply to this trinomial.

2 5 24x x Start with the first term in the trinomial,

2x ,this is the product of the first two terms of the

binomials. This means that the first term in each binomial is x.

Next, since the last term in the trinomial is –24, find two numbers that multiply to –24. Possible

choices are:

These two numbers are also multiplied by the first terms in the binomials (x and x). The sum

of these “inner” and “outer” products must add up to –5x, so choose the factors whose sum is –5.

This is the final answer. Notice that it is a product. The answer may also be written with the

order of the binomials switched: 3 8x x

Homework


2. What are the three types of factoring methods covered in Chapter 7 so far (Sections 7.1 – 7.3)?

3. When asked to factor an expression, how do you determine when to try factoring by “un-foiling”?




7. In your own words, describe how to “un-foil” a trinomial.

8. Factor each expression. Notice that in parts a–d, the absolute values of the constants in the two binomials are always the same,

but the signs are different depending both on the signs in the original trinomial and the middle coefficient of the original

trinomial.

a) 2 9 20x x

b) 2 9 20x x

c) 2 20x x

d) 2 20x x

e) In your own words, describe why the trinomial 2 9 20x x is prime (not factorable)?

f) In your own words, describe why the trinomial 2 20x x is prime (not factorable)?

Factors that multiply to –24 Sum of the two factors

1 24 1 24 23

1 24 1 24 23

2 12 2 12 10

2 12 2 12 10

3 8 3 8 5

3 8 3 8 5

4 6 4 6 2

4 6 4 6 2




9. 2 15 50x x

10. 2 12 11p p

11. 2 10 21y y

12. 2 6 8a a

13. 2 3 40m m

14. 2 2 63p p

15. 2 12 28x x

16. 2 3 18a a

17. 2 7 4y y

18. 2 4 32n n

19. 2 5 6x x

20. 2 17 52p p

21. 2 7 30y y

22. 2 2 15a a

23. 2 2 13 42c d cd

24. 2 2 2 24x y xy

25. 2 27 12p pq q

26. 2 2 30m n mn

27. 2 217 60x xy y

28. 2 220 3t t u u

29. 2 22x xy y

30. 22 26 72q q Hint: GCF first! Watch for this in all problems.

31. 25 15 60a a

32. 23 21 30n n

33. 2 24 32 48x y xy

34. 2 5 24p q pq q

35. 3 2 22 14 36c c d cd

36. 3 3 2 217 30x y x y xy

37. 6 314 33a a

38. 8 45 14z z



39. Factor each expression.

a) 2 10 25w w

b) 2 14 49y y

c) 2 6 9m m

d) In your own words, describe the pattern for the above three trinomials and their corresponding factored expressions.

e) Use this pattern to factor 2 1 12 16

x x .

40. Factor each expression.

a) 2 8 16p p

b) 2 12 36n n

c) 2 16 64a a

d) In your own words, describe the pattern for the above three trinomials and their corresponding factored expressions.

e) Use this pattern to factor 2 14

x x .


Section 7.4 — Factoring a Difference of Squares 8 Caspers

3 2 3 2x y x y

Section 7.4 Factoring a Difference of Squares

7.4 — Difference of Squares Practice Worksheet

Example:

Factor. Make sure the expression is factored completely. If the expression is not factorable, state that it is prime.

a) 2 29 4x y



There are two terms in the expression, check to see if the expression is a DIFFERENCE of squares. In other words, both the

first term and the last term must be perfect squares and the sign between MUST be a subtraction. There are other ways to

factor two terms; they will be covered in the next math course.

If the expression is a DIFFERENCE of squares, try factoring by “un-FOIL-ing”: find two binomials that multiply to this

binomial. Since there are only two terms (not three), the product of the two binomials follows a special pattern:

2 2a b a b a b

2 29 4x y Start with the first term in the trinomial, 29x ,this is a perfect square and must be the product of

the first two terms of the binomials; therefore, the first term in each binomial is 3x.

The last term in the trinomial, 24y , is also a perfect square and must be the product of the last

two terms of the binomials; therefore, the last term in each binomial (in absolute value) is 2y.

Notice that the “inner” and “outer” products add up to 0, since there is no middle term.


order of the binomials switched: 3 2 3 2x y x y . Remember, you may check any

factoring problem by multiplying your answer.

Homework



3. When asked to factor an expression, how do you determine when you are factoring a difference of squares?




7. In your own words, describe how to “un-foil” a difference of squares.

8. Is it possible to factor the expression 2 9x ? In your own words, explain why or why not.

9. Is it possible to factor the expression 24 16x ? In your own words, explain why or why not.

10. If an expression has two terms and it is not a difference of squares, can it be factored? In your own words, explain why or why

not.


11. 2 25x

12. 2 121a




Section 7.4 — Factoring a Difference of Squares 9 Caspers

13. 2 81p

14. 2 1y

15. 2 100m

16. 2 196k

17. 24 9c

18. 2 2x y

19. 281 1w

20. 249 16d

21. 236 n

22. 2 2 64s t

23. 2 2100 169w z

24. 4 144y

25. 6225 x

26. 2 14

p

27. 25 80a Hint: GCF first! Watch for this in all problems.

28. 23 363m

29. 3 196x x

30. 2 925

v

31. 22 98z

32. 225 225q

33. 216 4c

34. 4 2144 9a a

35. 28 8y

36. 4 16x

37. 412 12k


Section 7.5 — Factoring Trinomials with a non-1 Leading Coefficient 10 Caspers

2 5 3 1x x

Section 7.5 Factoring Trinomials with a non-1 Leading Coefficient

7.5 — Factoring Trinomials with a non-1 Leading Coefficient Worksheet

Example:

Factor. Make sure the expression is factored completely. If the expression is not factorable, state that it is prime.

a) 26 17 5x x



There are three terms in the expression, so try factoring by “un-FOIL-ing”: find two binomials that multiply to this trinomial.

Since the leading coefficient is not 1, more thought is required. If this is really challenging for you, ask your instructor about

the “ac-grouping method”.

26 17 5x x Start with the first term in the trinomial,

26x ,this is the product of the first two terms of the

binomials. This means that the first terms in each binomial are either: 2x and 3x OR 1x and 6x.

Notice that all terms are positive, so both signs in the binomials will be addition.

2 3x x OR 6x x

Next, since the last term in the trinomial is 5, find two numbers that multiply to 5. Since all signs

are positive, the only choices are: 1 and 5.

These two numbers are also multiplied by the first terms in the binomials (either: 2x and 3x OR

1x and 6x). The sum of these “inner” and “outer” products must add up to 17x.

Using 1 and 5 as the last terms with the above first terms, yields:

Although all of the resulting trinomials have the same first and last terms, each has a different

middle term. For this problem, choose the binomials that resulted in a middle term of 17x.


order of the binomials switched: 3 1 2 5x x . Remember, you may check any

factoring problem by multiplying your answer. Homework



3. When asked to factor an expression, how do you determine when to “un-FOIL” or use the a-c grouping method?




7. Choose one of the following:

In your own words, describe how to “un-FOIL” a trinomial with a non-1 leading coefficient

OR

In your own words, describe how to factor a trinomial with a non-1 leading coefficient using the a-c grouping method.

Binomial Choices “FOIL” Simplified Product

2 1 3 5x x 26 10 3 5x x x 26 13 5x x

2 5 3 1x x 26 2 15 5x x x 26 17 5x x

1 6 5x x 26 5 6 5x x x 26 11 5x x

5 6 1x x 26 30 5x x x 26 31 5x x




Section 7.5 — Factoring Trinomials with a non-1 Leading Coefficient 11 Caspers

8. Is it possible to factor the expression 22 3 1x x even though there are no factors of 1 that add up to 3? Why or why not?

9. Factor each expression. Notice: the absolute values of the constants in the two binomials are always the same, but the problems

are different depending on signs, the value of the coefficient in the trinomial, and the placement of the constants in the binomials.

a) 22 15 7x x

b) 22 15 7x x

c) 22 9 7x x

d) 22 9 7x x

e) 22 13 7x x

f) 22 13 7x x

g) 22 5 7x x

h) 22 5 7x x


10. 22 5 3y y

11. 23 16 5m m

12. 25 17 14x x

13. 26 35 11a a

14. 26 17 11p p

15. 24 13 12d d

16. 24 4 15x x

17. 23 2 8k k

18. 28 18 9q q

19. 29 48 64c c

20. 22 21x x

21. 210 159 16y y

22. 29 34 8n n

23. 236 60 25a a

24. 221 10 16m m

25. 2 22 13 15x xy y

26. 2 29 6a ad d

27. 2 212 17 5p q pq

28. 2 215 8 12c d cd

29. 2500 300 45n n Hint: GCF first! Watch for this in all problems.

30. 3 220 9 18q q q

31. 3 228 54 18k k k

32. 2 236 186 240x xy y


Section 7.6 — General Factoring 12 Caspers

Section 7.6 General Factoring

7.6 — General Factoring Worksheet

Examples:


a) 44 4x


For this expression, there is a GCF for all terms, so factor it out. 44 4x 44 1x

Next, check to see if the remaining expression (without the GCF) can be factored further.

This expression has two terms and is a difference of squares.

44 1x 2 24 1 1x x

Now check to see if any of the factors can be factored further.

In this problem, 2 1x cannot be factored further, but 2 1x can be factored further.

2 24 1 1x x 24 1 1 1x x x

The final answer is: 24 1 1 1x x x . Factors may be written in any order.

b) 3 2 215 27 6x x y xy


For this expression, there is a GCF for all terms, so factor it out. 3 2 215 27 6x x y xy 2 23 5 9 2x x xy y

Next, check to see if the remaining expression (without the GCF) can be factored further.

This expression has three terms, so try to “un-FOIL”.

2 23 5 9 2x x xy y 3 5 2x x y x y

The final answer is: 3 5 2x x y x y . Factors may be written in any order.

Homework


2. What are the all of the types of factoring methods covered in Chapter 7 so far (Sections 7.1 – 7.6)?







Section 7.6 — General Factoring 13 Caspers

Factor. Make sure each expression is factored completely. If the expression is not factorable, state that it is prime. Show all steps.

Check ALL answers with answers provided for this workbook.

6. 2 100t

7. 24 12x x

8. 2 30y y

9. 3 23 7 21p p p

10. 218 3 10m m

11. 23 45 162a a

12. 2 9z

13. 25 125x

14. 3 212 50 18y y y

15. 4 3 25 5 3 3n n n n

16. 2675 3q

17. 2 144t t

18. 3 25 4 20a a a

19. 2 24 32 60c cd d

20. 48 8x

21. 4 212 13z z

22. 3 264 176 121m m m

23. 2 281 16p q

24. 3 3 2 214 29 12a c a c ac

25. 5 4 3 218 90 48 240y y y y

26. 22 300 1n n

27. 7 6 510 90 80k k k

28. 2 22 72 36wz z w

29. 4 4x y

30. 3 230 123 135m m m

31. 4 22 96 98p p

32. 3 2160 60 40 15a a a

33. 3 2 2 318 112 24x y x y xy


Section 7.7 — Solving Quadratic Equations by Factoring 14 Caspers

12

2 1 0

2 1

x

x

x

2 0

2

x

x

Section 7.7 Solving Quadratic Equations by Factoring

Example:

Solve the equation. Give simplified answers.

a) 2 3 2x x

To solve a quadratic equation by factoring, first multiply/distribute and combine like-terms, in order to get the equation into

standard form: 2 0ax bx c

2 3 2x x 22 3 2x x

22 3 2 0x x

Once the equation is set equal to zero and all like-terms are combined, factor.

22 3 2 0x x 2 1 2 0x x

Use the Zero Product Rule to find the values for x that satisfy the equation, by setting each factor equal to zero and solve.

2 1 2 0x x AND

The solutions are ½ and –2.

Homework

1. In your own words, describe a linear equation. How does one recognize that an equation is linear? Give two examples of linear

equations.

2. In your own words, describe a quadratic equation. How does one recognize that an equation is quadratic? Give two examples of

quadratic equations.

3. In your own words, describe how solve a quadratic equation by factoring.

Solve each equation. Show a check for every third problem. Give simplified answers.

4. 2 4 60 0x x

5. 2 3 28p p

6. 3 12 0y y

7. 2 56 15q q

8. 22 7 5 0n n

9. 23 3 60a a

10. 22 4 0x x

11. 2 17 82 10m m

12. 23 15 40 8k k k

13. 2 25 10y y

14. 2 8 1a

15. 2 5p p


Section 7.7 — Solving Quadratic Equations by Factoring 15 Caspers

16. 2 9 12 44n n

17. 13 6 0x x

18. 7 3 0q q

19. 26 100 56 20k k

20. 212

2 20 9 0m m

21. 2 512 2

7x x

22. 2 274

1 0y y

23. 2 16 60 12p p

24. 2

11 0a

25. 2

3 2 18z

26. 2 24 24k k

27. 5 2 9 15 2x x x

28. 4 1 7 1n n n

29. 2

4 3 40 2y y

30. 7 2 6q q


Section 7.8 — General Application Problems Involving Quadratic Equations 16 Caspers

22

2

2 2

2

2

3 35

2 2 1 6 9

2 2 2 2 1 6 9

4 4 4 4 1 6 9

5 8 4 1 6 9

5 8 1 6 5 0

5 3 3 5 0

5 3 3 0 O R 5 0

5 3 3 5

x x

x x x

x x x x

x x

x x

x x

x x

x x

x

Section 7.8 General Application Problems Involving Quadratic Equations

7.8 — General Application Problems Involving Quadratic Equations Worksheet

Example:

For each problem below, define variables, write an equation, solve the equation, and answer the question(s). Give your answer(s) in

simplest form with the correct units when appropriate.

a) One leg of a right triangle is two inches more than twice the other leg. The hypotenuse is 13 inches. Find the lengths of

the three sides of the triangle.

The “other” leg of the right triangle: x

“One” leg of the right triangle: 2 2x Define what the problem asks to find.

The hypotenuse of the right triangle: 1 3

The Pythagorean Theorem is about the lengths of the sides of a right triangle. Use the appropriate geometric fact to write 2 2 2a b c an equation.

the square of the length of the “other” leg + the square of the length of the “one” leg = the square of the hypotenuse

22 22 2 13x x

Solve the equation. “FOIL” when squaring

the binomial. Combine like terms. This is a

quadratic equation, so get a zero on one

side and then factor.

The value for x cannot be negative for this

problem, since it represents the length of a

leg of a triangle.

The “other” leg is 5 inches, the “one” leg is 2 5 2 = 12 inches, and the hypotenuse is 13 inches.

Homework

For each problem below, define variables, write an equation, solve the equation, and answer the question(s). Give your answer(s) in

simplest form with the correct units when appropriate.

1. Determine the value of x.


x

5 cm

12 cm

5 feet

x

3 feet




Section 7.8 — General Application Problems Involving Quadratic Equations 17 Caspers


4. One leg of a right triangle is two inches less than the other leg. The hypotenuse is 10 inches. Find the lengths of the three

sides of the triangle.

5. A ladder is leaning against a house. If the base of the ladder is 5 feet from the house and the ladder reaches 12 feet high on the

house, how long is the ladder?

6. The product of two consecutive positive even integers is 288. Find the two integers.

7. One integer is three less than twice the other integer. Their product is 35. Find the two integers.

8. One number is four more than another number. Their product is 96. Find the two numbers. Two pairs of numbers satisfy these conditions.

9. One number is two more than twice another number. Their product is 12. Find the two numbers. Two pairs of numbers satisfy these conditions.

10. The area of a rectangle is 88 square inches. Determine the length and width if the length is five inches less than twice the

width.

x 26 inches

24 inches

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