1 Section 5.4 Factoring FACTORING Greatest Common Factor, Factor by Grouping, Factoring Trinomials, Difference of Squares, Perfect Square Trinomial, Sum

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Section 5.4 FactoringFACTORING

Greatest Common Factor, Factor by Grouping, Factoring Trinomials,

Difference of Squares, Perfect Square Trinomial, Sum & Difference of Cubes

2

Factoring—define factored form

Factor means to write a quantity as a multiplication problem

a product of the factors. Factored forms of 18 are:

118, 2 9, 3 6

3

Factoring: The Greatest Common Factor

To find the greatest common factor of a list of numbers:

1. Write each number in prime factored form2. Choose the least amount of each prime that

occurs in each number

3. Multiply them together Find the GCF of 24 & 36

24 2 2 2 3

36 2 2 3 3

2 2 3 12

4


To find the greatest common factor of a list of variable terms:

1. Choose the variables common to each term.

2. Choose the smallest exponent of each common variable.

3. Multiply the variables. Find the GCF of:

4 2 7 3 5 2 6 2, , , x y x y x y x y x y

5


To factor out the greatest common factor of a polynomial:

1. Choose the greatest common factor for the coefficients.

2. Choose the greatest common factor for the variable parts.

3. Multiply the factors.

6


Factor

each of the

following

by factoring

out the

greatest

common

factor:

5x + 5 =

4ab + 10a2 =

8p4q3 + 6p3q2 =

2y + 4y2 + 16y3 =

3x(y + 2) -1(y + 2) =

7


The answers are :

2

4 3 3 2 3 2

2 3 2

5 5 5 1

4 10 2 2 5

8 6 2 4 3

2 4 16 2 1 2 8

3 2 1 2 2 3 1

x x

ab a a b a

p q p q p q pq

y y y y y y

x y y y x

8

Factoring: by Grouping

Often used when factoring four terms.

1. Organize the terms in two groups of two terms.

2. Factor out the greatest common factor from each group of two terms.

3. Factor out the common binomial factor from the two groups.

Rearranging the terms may be necessary.

9


Factor by grouping:1. 2 groups of 2 terms

2. Factor out the GCF

from each group of 2 terms

3. Factor out the

common binomial factor

2 8 3 12

2 4 3 4

4 2 3

mn n m

n m m

m n

10


Factor

by

grouping

2 2

2 2

2 2

6 20 15 8

2 3 10 ????????

rearrange the terms and try again

6 8 15 20

2 3 4 5 3 4

3 4 2 5

y w yw yw

y w

y yw yw w

y y w w y w

y w y w

11

Factoring Trinomials—with a coefficient of 1 for the squared term

Factor:

1. List the factors of 20:

2. Select the pairs from which 12

may be obtained

3. Write the two

binomial factors:

4. Check using FOIL:

2 12 20x y 20

1 20

2 10

4 5

2

2

10 2

2 10 20

12 20

x x

x x x

x x

12

Factoring Trinomials TIP

If the last term of the trinomial is positive and the middle sign is positive, both binomials will have the same “middle” sign as the second term.

2 12 20

10 2

x x

x x

13


If the last term of the trinomial is positive and the middle sign is negative, both binomials will have the same “middle” sign as the second term.

2 12 20

10 2

x x

x x

14

Factoring Trinomials—with a coefficient of 1 for the squared term

Factor 1. List the factors of 222. Select the pair from

which –9 may be obtained3. Write the two

binomial factors:

4. Check using FOIL:

2 9 22a a 22

1 2

1

2

2 1

2

2

11 2

2 11 22

9 22

x x

x x x

x x

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If the last term of the trinomial is negative, both binomials will have one plus and one minus “middle” sign.

2 9 22

11 2

x x

x x

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Factoring Trinomials—primes

A PRIME POLYNOMIAL cannot be factored using only integer factors.

Factor :The factors of 5: 1 and 5.Since –2 cannot be obtained from 1 and

5, the polynomial is prime.

2 2 5x x

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Factoring Trinomials—2 variables

Factor: The factors of 8 are: 1,8 & 2,4, & -1,-8 & -2, -4 Choose the pairs from which –6 can be obtained: 2 & 4 Use y in the first position and z in the second position Write the two binomialfactors and check your answer

2 2

2 2

4 2

2 4 8

6 8

y z y z

y yz yz z

y yz z

2 26 8y yz z

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Factoring Trinomials—with a GCF

If there is a greatest common factor? If yes, factor it out first.

4 3 2

2 2

2

2

Factor: 3 15 18

3 5 6 factors of 6: 1 6 & 2 3

3

Choose 2 3

3 2 3

z z z

z z z

z z z

z z z

19

Factoring Trinomials—always check your factored form

Always check your answer with multiplication of the factors.

The check:

2

2 2

2 2

4 3 3

3 2 3

3 3 2 6

3 5 6

3 15 18

z z z

z z z z

z z z

z z z

20

Factoring Trinomials—when the coefficient is not 1 on the squared term

2

2

Factor: 3 4 1 1. Multiply 3 1 3

2. List the factors of 3: 1 3

3. Rewrite the middle term of the trinomial using

the factors of 3.

3 3 1

x x

x x x

4. Factor by grouping.

3 1 1 3 1 1 3 1x x x x x

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Factoring Trinomials---use grouping

2

2

Factor: 6 1 1. Multiply 6 1 6

2. List the factors of 6: 1 6 & 2 3

3. Rewrite the middle term of the trinomial using the

factors of 6 which add to be 1 ie. 2 3.

6 3 2 1

4.

x x

x x x

Factor by grouping.

3 2 1 1 2 1 2 1 3 1x x x x x

22

Factoring Trinomials---use grouping

2

2

Factor: 12 5 2 1. Multiply 12 2 24

2. List the factors of 24: 1 24; 2 12; 3 8; & 4 6

3. Rewrite the middle term of the trinomial using

the factors of 24 which add to be 5 ie. 3 8.

12 8

x x

x x

3 2

4. Factor by grouping.

4 3 2 1 3 2 3 2 4 1

x

x x x x x

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Factoring Trinomials---use FOIL and Trial and Error

2Factor: 6 19 10

There will be two binomial factors.

Both middle signs will be positive.

The factors of 6 are 1 6 and 2 3.


(Continued o

x x

n next screen.)

24


Think FOIL and focus on the outers and inners

while using trial and error to position the

factors correctly. 2 5 3 2

inners: 15

x x

x

2

outers: 4

Always check your answer.

2 5 3 2 6 19 10

x

x x x x

25


2Factor: 10 23 12


Both middle signs will be negative.


The factors of 12 are 1 12, 2 6, and 3 4.

(C

x x

ontinued on next screen.)

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factors correctly. 2 3 5 4

inners: 15

x x

x

2

outers: 8


2 3 5 4 10 23 12

x

x x x x

27


2Factor: 5 13 6


One middle sign will be negative.

One middle sign will be positive.

The factors of 5 are 1 5.

The factors of 6 are 1 6

x x

, and 2 3.

(Continued on next screen.)

28




factors correctly. 5 2 3

inners: 2

x x

x

2

outers: 15


5 2 3 5 13 6

x

x x x x

29

Factoring Trinomials---with a negative GCF

Is the squared term negative? If yes, factor our a negative GCF.

2

2

Factor: 4 2 30

2 2 15

2 2 5 3

The sum of inners: 5 and outers: 6 is

a a

a a a

a a a

a a a

30

Special Factoring—difference of 2 squares

The following must be true:1. There must be only two terms in the

polynomial.

2. Both terms must be perfect squares.

3. There must be a “minus” sign between the two terms.

31


The following pattern holds true for the difference of 2 squares:

2 2x y x y x y

32


The pattern:

2 2x y x y x y

2

2 2

22

Factor: 25

is a perfect square since

25 is a perfect square since 25 5

Use the pattern letting 5.

5 5 5

x

x x x

y

x x x

33


The pattern:

2 2x y x y x y

2 2

2 2

2 2

2 2

Factor:



Use the pattern letting & .

a b

a a a

b b b

x a y b

a b a b a b

34


The pattern:

2 2x y x y x y

2 2

2 2

2 2

2 2

Factor: 9 4



Use the pattern letting 3 & 2 .

3 2 3 2 3 2

x y

x x x

y y y

x x y y

x y x y x y

35


The pattern:

2 2x y x y x y

2 2Factor: 9 4

Although both terms are perfect squares,

the pattern does not apply because the

problem is a sum and not a difference.

x y

36

Special Factoring—perfect square trinomial

A perfect square trinomial is a trinomial that is the square of a binomial.

2

2 2 2 2

2 2 2

2 2 2 4 4

x y x y x y

x xy xy y x xy y

2 24 4 is a perfect square trinomial because:x xy y

37


The first and third terms are perfect squares. AND the middle term is twice the product of the

square roots of the first and third terms TEST THE MIDDLE TERM:

2 24 4 is a perfect square trinomial because:x xy y

2 24 2 2 2 4x x y y x y xy

38


The patterns for a perfect square trinomial are:

22 2

22 2

2

2

x xy y x y x y x y

x xy y x y x y x y

39


Factor the following using the perfect square trinomial pattern:

22

2

16 56 49 4 7

The first and third terms are perfect squares.

16 4 49 7

The middle term is twice the product of the

square root of the first and third terms.

2 4 7 56

a a a

a a

a a

40


Factor the following using the perfect square trinomial pattern:

22 2

2 2

22 121 11

The first and third terms are perfect squares.

121 11

The middle term is twice the product of the

square root of the first and third terms.

2 11 22

x xz z x z

x x z z

x z xz

41

Special Factoring—difference of two cubes

Factor using the pattern.

3 3 2 2x y x y x xy y

3 3

3 3 33

3 3

2 2

2 2

27 8

Both terms must be perfect cubes.

27 3 ; 8 2

3 2 Let 3 , Let 2

3 2 3 3 2 2

3 2 9 6 4

x y

x x y y

x y x x y y

x y x x y y

x y x xy y

42

Special Factoring—sum of two cubes

Factor using the pattern.

3 3 2 2x y x y x xy y

3 3

3 3 33

3 3

2 2

2 2

1000 125

Both terms must be perfect cubes.

1000 10 ; 125 5

10 5 Let 10 , Let 5

10 5 10 10 5 5

10 5 100 50 25

x y

x x y y

x y x x y y

x y x x y y

x y x xy y

43

Solving quadratic equation with factoring

A quadratic equation has a “squared” term.

2a b c 0x x

44

ZERO FACTOR PROPERTY

To Factor a Quadratic,

Apply the Zero-Factor Property.

If a and b are real numbers and if ab = 0, then a = 0 or b = 0.

45

Solving quadratic equations with factoring—Zero-Factor Property

Solve the equation:

(x + 2)(x - 8) = 0.

Apply the zero-factor property:

(x + 2) = 0 or (x – 8) = 0

x = -2 or x = 8

46

Solving quadratic equations with factoring—Zero-Factor Property

There are two answers for x:

-2 and 8.Check by substituting the values

calculated for x into the original equation.

(x + 2)(x - 8) = 0.

(-2 + 2)(-2 – 8) = 0 (8 + 2)(8 – 8) = 0

0 = 0 0 = 0

47

Solving quadratic equations with factoring—Standard Form

To solve a quadratic equation,

1. Write the equation in standard form. (Solve the equation for 0.)

2 26 8 6 8 0x x x x

48

Solving quadratic equations with factoring


2. Factor the quadratic expression.

2 6 8 0

2 4 0

x x

x x

49



3. Apply the Zero-Factor Property

2 6 8 0

2 4 0

2 0 4 0

2 4

x x

x x

x x

x x

50



4. Check your answers

2

2 2

6 8 0 2, or 4

2 6 2 8 0 4 6 4 8 0

4 12 8 0 16 24 8 08 8 0 8 8 0

0 0 0 0

x x x x

Documents

1 Section 5.4 Factoring FACTORING Greatest Common Factor, Factor by Grouping, Factoring Trinomials, Difference of Squares, Perfect Square Trinomial, Sum