factoring polynomials (part 1) – gcf and un-f-o-i-l §5 – 1:...

Algebra I Name: page 1 of 19

CLASS NOTES: §5 – 1 thru §5 – 3, §5 – 7, §5 – 8 Factoring Polynomials (Part 1) – GCF and un-F-O-I-L

§5 – 1: Factoring Numbers, Greatest Common Factor (GCF) This unit is about FACTORING.

A factor is something that is multiplied. A factor can be a number, a variable, a monomial or even a polynomial.

To factor means to write something as a product of its factors.

A number or expression is called prime if its only factors are 1 and itself. (Examples of prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, …)

The greatest common factor (GCF) of two numbers or expressions is the

largest number that is a factor of both of the numbers or expressions. EX 1 List the factor pairs for each number. (a) 20 (b) -12

�

1( ) 20( ) −1( ) −20( )2( ) 10( ) −2( ) −10( )4( ) 5( ) −4( ) −5( )

�

−1( ) 12( ) 1( ) −12( )−2( ) 6( ) 2( ) −6( )−3( ) 4( ) 3( ) −4( )

(c) 32 (d) 48 (e) 56 (f) -70 (g) -80 (h) 100

Remember…

�

factor × factor = product


EX 2 Find the GCF of each pair of numbers (a)

�

72 , 108

GCF is…

�

9 • 4 = 36

�

9 72 , 108

4 8 , 12

2 , 3

(b)

�

32 , 80 (c)

�

300, 96 (d)

�

360, 168 (e)

�

48 , 80 , 112

(f)

�

8ax 2 , 12a 2 x

This example has variables! Before we can factor expressions with variables we need to learn how to divide with variables.


§5 – 2: Dividing Monomials Remember our rules for multiplying monomials…

�

x a • x b = x a + b EX:

�

x 2⎛ ⎝ ⎜ ⎞

⎠ ⎟ x 5⎛ ⎝ ⎜ ⎞

⎠ ⎟ = x 2 + 5 = x 7

The rules for dividing monomials are similar… ** if the power in the numerator is larger…

EX:

�

x 5

x 3= x 5−3 = x 2

** if the power in the denominator is larger…

EX:

�

x 2

x 5=

1

x 5−2=

1

x 3

There are more x’s multiplied on top. How many more??? _______

�

x a

x b= x a −b

There are more x’s multiplied on bottom. How many more??? _______

�

x a

x b=

1

x b −a


EX 1 Simplify each fraction using the Laws of Exponents.

(a)

�

x 7

x 4 (b)

�

a 3

a 8

(c)

�

n 5

n 5 (d)

�

x 9

x 5

(e)

�

x

x (f)

�

16xy

8x

(g)

�

15n

5n (h)

�

9x 4

12 x

(i)

�

18x 4

12 x 3y ⇐

(j)

�

−7x 2y 3

28xy (k)

�

−12 x 2y

−20xy 2

To make these less confusing, break them into parts…

�

=18 x 4

12 x 3 y=

3x

2 y=

3x

2y

↑ ↑ ↑ 6 one one goes x y is into left left on both on the top bottom


EX 2 Simplify each fractions using the Laws of Exponents. (Hint: First simplify the numerator and denominator separately.)

(a)

�

4x 3⎛ ⎝ ⎜ ⎞

⎠ ⎟

2

2 x 3⎛ ⎝ ⎜ ⎞

⎠ ⎟

3

�

42 x 3•2

2 3 x 3•3=

16x 6

8x 9=

2

x 3

(b)

�

xy( ) 7

xy 7 (c)

�

3x 2y⎛ ⎝ ⎜ ⎞

⎠ ⎟

3

15x 2y 2

(d)

�

−x 5⎛ ⎝ ⎜ ⎞

⎠ ⎟

2

−x 2⎛ ⎝ ⎜ ⎞

⎠ ⎟

3 (e)

�

2 x 2⎛ ⎝ ⎜ ⎞

⎠ ⎟

3

2 x 3⎛ ⎝ ⎜ ⎞

⎠ ⎟

2

EX 3 Find the greatest common factor (GCF) of each set of numbers.

(a)

�

8ax 2 , 12a 2 x

The GCF is…

�

4 • a • x = 4ax

�

a 8ax 2 , 12a 2 x

x 8x 2 , 12ax

4 8x , 12a

2 x , 3a

Remember…

�

x a⎛ ⎝ ⎜ ⎞

⎠ ⎟

b= x a •b

nx a⎛ ⎝ ⎜ ⎞

⎠ ⎟

b= n b x a •b


EX 4 Find the GCF of each set of numbers

(a)

�

48x 2y 2 , 60xy 3z

(b)

�

72 x 3yz 3 , 120x 2 z 5

(c)

�

32 x 3y 3 , 120xy 4

(d)

�

132 xy 3 , 44x 2 , 99xy 4 EX 5 Find the missing factor. (a)

�

48 = 12( ) ___( ) ⇐ The question is “12 times what number is 48?”

Think… what is

�

48 ÷ 12 ? The answer is 4 !

(b)

�

18x 9 = 6x 4⎛ ⎝ ⎜ ⎞

⎠ ⎟ _____( )

(c)

�

16x 4y 3 = 8x 2y⎛ ⎝ ⎜ ⎞

⎠ ⎟ ______( )

Remember…

�

factor × factor = product


§5 – 3: Factoring the GCF using Distribution Multiplying and Factoring are opposites. Multiplying: Distribution Factoring: Backward Distribution

�

5 x − 7( ) ⇒ 5x − 35

�

5x − 35 ⇒ 5 x − 7( ) factor factor product product factor factor

�

5x − 35 ⇒ 5 x − 7( )

In order to factor, we must understand how to divide! EX 1 Divide. (Split into separate fractions.)

�

(a) 5x − 35

5=

5x

5−

35

5

= x − 7

(b)

�

12 xy − 18y

6y=

12 xy

6y−

18y

6y

This is the GCF

This is the answer to…

�

5x − 35

5


EX 2 Divide. (Split into separate fractions.)

(a)

�

9x + 12

3=

9x

3+

12

3

(b)

�

16x + 20y

4=

+

(c)

�

−27x 3 + 18x 2 − 36x

−9 x=

(d)

�

9x 3y + 3x 2y 2 + 3xy

3xy=


Now that we know how to divide, we can factor !! EX 3 Factor out the greatest common factor (GCF) using distribution

(backwards).

(a)

�

5x 2 + 10x

Step 1: Find GCF.

�

5 5x 2 + 10x

x x 2 + 2 x

x + 2

GCF is…

�

5 • x = 5x

Step 2: Factor out the GCF using distribution (backwards).

�

5x x + 2( ) ⇐ Check your answer by distributing.

(b)

�

6x − 18 Step 1: Find GCF. Step 2: Factor out the GCF using distribution (backwards).

Why did we learn to divide??? Because…

�

5x 2 + 10x

5x=

5x 2

5x+

10x

5x

= x + 2

GCF

This is dividing!


EX 4 Factor out the GCF from each expression using the DISTRIBUTIVE

PROPERTY.

(a)

�

4x 5 − 6x 3 + 14x Step 1: Find the GCF. Step 2: Divide by the GCF. Factored form is…

(b)

�

10x 3 − 25x 2 Step 1: Find the GCF. Step 2: Divide by the GCF. Factored form is…

(c)

�

45x 3 − 15x 2 − 15

(d)

�

35x 3 − 7x 2 − 14x


EX 5 Factor out the GCF from each expression using the DISTRIBUTIVE PROPERTY.

(a)

�

3xy 2 − 15x 2y + 12 xy NEGATIVE GCF: Sometimes you may find it useful to be able to factor out a negative number from a polynomial. Here is an example: EX 6 Factor out a positive GCF from this expression using the distributive

property.

�

24 + 6x − 2 x 2 = 2 12 + 3x − x 2⎛ ⎝ ⎜ ⎞

⎠ ⎟

Factor out a negative GCF from this expression using the distributive property.

�

24 + 6x − 2 x 2 = −2 −12 − 3x + x 2⎛ ⎝ ⎜ ⎞

⎠ ⎟

= −2 x 2 − 3x − 12⎛ ⎝ ⎜ ⎞

⎠ ⎟

BOTH ARE CORRECT! EX 7 Factor out a NEGATIVE GCF from each expression using the distributive

property.

(a)

�

5 + 10x − 30x 2

(b)

�

xy 2 − x 2y


§5 – 7: Factoring Quadratic Polynomials

�

x 2 + bx + c , c is positive Quadratic Polynomials - Polynomials that are in the form …

�

ax 2 + bx + c

where a, b and c are any number. (a is not zero)

�

x + 2( ) and x + 6( ) are the factors of the quadratic polynomial

�

x 2 + 8x + 12

because when we use F-O-I-L to multiply

�

x + 2( ) x + 6( ), we get

�

x 2 + 8x + 12.

Multiplying Using F-O-I-L Factoring (Un F-O-I-L)

�

x + 2( ) x + 6( ) ⇒ x 2 + 8x + 12

�

x 2 + 8x + 12 ⇒ x + 2( ) x + 6( ) factor factor product product factor factor

EX Factor

�

x 2 + 14x + 24

�

↑ ↑ ↑ F O + I L x • x __ • __

⇒ If the Firsts are

�

x • x , then the factors are…

�

x + ____( ) x + ____( ) ⇒ The Lasts must MULTIPLY to equal 24 and ADD to equal 14

�

____ • ____ = 24____ + ____ = 14

⇒ Therefore,

�

x 2 + 14x + 24 = x + 2( ) x + 12( )

Factoring a quadratic polynomial is like “un-F-O-I-L”

2 and 12 work!


EX 1 Factor the following quadratic polynomials.

(a)

�

x 2 + 9x + 20

�

↑ ↑ ↑ F O + I L x • x __ • __ __ + __ = 9


�


�

x + ____( ) x + ____( )

⇒ The Lasts must MULTIPLY to equal 20 and ADD to equal 9

Answer:

(b)

�

x 2 + 11x + 24

�

↑ ↑ ↑ F O + I L x • x __ • __ __ + __ = 11


�


�

x + ____( ) x + ____( )

⇒ The Lasts must MULTIPLY to equal 24 and ADD to equal 11

Answer:

(c)

�

x 2 − 11x + 24

�

↑ ↑ ↑ F O + I L x • x __ • __ __ + __ = −11

Answer:

Hint: Both numbers will be NEGATIVE!



(a)

�

x 2 + 14x + 40

�

↑ ↑ ↑ F O + I L x • x __ • __ __ + __ = 14

Answer:

(b)

�

x 2 − 11x + 18 (c)

�

x 2 + 9x + 20

(d)

�

x 2 − 10x + 14 (e)

�

x 2 − 9x + 8

(f)

�

x 2 + 16x + 48 (g)

�

x 2 − 17x + 42


EX 3 Factor.

(a)

�

x 2 − 10xy + 21y 2

x 2 − 10x !!+ 21!!!!= x − 3( ) x − 7( )

x 2 − 10xy + 21y 2 = x − ___( ) x − ___( )

(b)

�

x 2 − 7xy + 10y 2

(c)

�

x 2 + 6xy + 8y 2

(d)

�

x 2 + 19xy + 34y 2

−3y( ) −7y( ) = 21y 2


§5 – 8: Factoring Quadratic Polynomials

�

x 2 + bx + c , c is negative

EX Factor

�

x 2 + 8x − 20

�

↑ ↑ ↑ F O + I L x • x __ • __ __ + __ = 8


�


�

x + ____( ) x − ____( )

⇒ List all the possible factor pairs of -20

�

1( ) −20( )−1( ) 20( )2( ) −10( )−2( ) 10( )4( ) −5( )−4( ) 5( )


(a)

�

x 2 − x − 20

�

↑ ↑ ↑ F O + I L x • x __ • __ __ + __ = −1


�


�

x + ____( ) x − ____( )


Answer:

�

x + ____( ) x − ____( )

The Lasts MULTIPLY to equal NEGATIVE 20

• one factor is POSITIVE and • one factor is NEGATIVE

This pair ADDS to 8 so the factors are:

�

x − 2( ) x + 10( )



(a)

�

x 2 + 29x − 30

�

↑ ↑ ↑ F O + I L x • x __ • __ __ + __ = 29


�


�

x + ____( ) x − ____( )


Answer:

�

x + ____( ) x − ____( )

(b)

�

x 2 − 7x − 30

�

↑ ↑ ↑ F O + I L x • x __ • __ __ + __ = −7


Answer:

(c)

�

x 2 − 2 x − 35 List factor pairs of -35

�

↑ ↑ ↑ F O + I L x • x __ • __ __ + __ = −2



(a)

�

x 2 + 10x − 24 List factor pairs of -24

�

↑ ↑ ↑ F O + I L x • x __ • __ __ + __ = 10

Answer:

(b)

�

x 2 + 5x − 24 (c)

�

x 2 + 4x − 5

(d)

�

x 2 − 7x − 18 (e)

�

x 2 − 32 x − 33

(f)

�

x 2 + x − 12 (g)

�

x 2 − 14x − 32


EX 4 Factor.

(a)

�

x 2 − 3xy + 18y 2

x 2 − 3x !!− 18!!!!= x − 6( ) x + 3( )

x 2 − 3xy − 18y 2 = x − ___( ) x + ___( )

(b)

�

x 2 + 4xy − 45y 2

(c)

�

x 2 − 4xy − 12y 2

(d)

�

x 2 − 34xy − 72y 2

−6y( ) 3y( ) = −18y 2

factoring polynomials (part 1) – gcf and un-f-o-i-l §5 – 1:...

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