Factoring Trinomials

Section 6.2

MATH 116-460 Mr. Keltner

Monomials and more monomials

 A

monomial

is an algebraic expression that is either a number, a variable, or the product of a number and a variable.

  A

binomial

, such as

x - 7,

monomials.

is the sum of two A

trinomial

, such as

x

2 three monomials.

+ 8

x

+15, is the sum of  A

polynomial

is the group that all of these expressions belong to.

 Monomials, binomials, and trinomials are just specific types of polynomials.

Factoring Quadratic Expressions

 When we worked with multiplying expressions such as 2

x

and (

x

as a sum, like 2

x

2 + 3), we wrote the product + 6

x

.

 This used the Distributive Property.



Factoring

reverses the process, allowing us to write the sum as a product.

 To factor an expression containing two or more terms, try factoring out the

greatest common factor

(GCF).

Example 1

 Factor each quadratic expression, by finding the greatest common factor (GCF) of the expression.

 27

c

2 - 18

c

 5

z

(2

z

+ 1) – 2(2

z

+ 1)

Factoring algebraically

 Factor

x

2 + 3x – 10

.

 Look for a pattern.

 We will notice that the sums and products in the x-term and constant term are related to the

factors

of the last term in the factored expression in the form

(x -b)(x - c)

.

 This observation gives us a rule for factoring quadratic expressions of the form

x 2

+ bx + c

.

Factoring x

2

+ bx + c

 To factor an expression of the form

ax 2

+ bx + c

where

a = 1

, look for integers

j

that

j

•

k

=

c

and

j

+

k

=

b

.

and

k

such  Then factor the expression.

x 2

+ bx + c = x

2

+ (j + k)x + ( j • k) = (x + j) (x + k)

Example 2

 Factor each quadratic expression.



x

2 + 12

x

+ 27 

n

2 - 4

n

- 12 

y

2 + 10

y

- 24

If there’s another number in front…

 Check for a monomial that can be factored out of each term of the expression.

 Example 3: Factor the expressions  3

n

3 - 18

n

2 + 24

n



y

5 + 3

y

4 - 18

y

3

Don’t get overwhelmed by seeing several variables

 By keeping the FOIL method in mind and knowing where the last term comes from, factoring trinomials with more than one variable is easier than it looks.

 It is always a good idea to look for a monomial factor.



Example 4

: Factor the expressions 

x

2 + 9

xy

+ 20

y

2 

x

3 + 4

x

2

y

- 21

xy

2

Factoring ax

2 + bx + c

when a ≠1: Trial and Error

  In the case where our leading coefficient is NOT 1 and we cannot factor a monomial out of the expression,consider  The factors of the

ax 2

term  The factors of the

c

term Keep in mind some behaviors of odd and even numbers: 

Even + Even = Even = Odd + Odd

 

Even • Even = Even = Even • Odd

 

Odd • Odd = Odd



Example 5: Factoring by Trial and Error

 Factor each expression and check by using the FOIL method.

 6

t

2 - 19

t

+ 10  56

n

4 - 70

n

3 + 21

n

2  6

x

3 - 28

x

2

y

- 48

xy

2

Factoring by Grouping

 To factor a trinomial of the form

ax 2 c

, where

a≠1

, by using

grouping: + bx +

 Look for a monomial GCF in all the terms. Factor it out, is possible.

 Find two factors of

ac

the numbers to

b

.

a

(are divisible by one of or

c

) which happen to add up  Write the polynomial with four terms so that the

bx

term is written as the sum of two like terms with coefficients that we used in step 2.

 Factor by grouping!

Factoring by Grouping: Your turn

 Factor each expression by grouping.

 10

x 2

- 19

x

+ 6  36

x 4 y

+ 3

x 3 y

- 60

x 2 y

Using substitution for something better to look at

Substitution

 Some rather ugly polynomials are actually in the form

ax 2 + bx + c

and can be factored by using a method called

substitution

.

 This takes advantage of taking a messy

part

of the expression and replacing it with a single variable.

Example 7: Factoring by Substitution

 Factor each expression by using substitution:  24(

t

+ 2) 2 - 22(

t

+ 2) + 3  15

n

8 - 19

n

4 + 6

Assessment

Pgs. 403-404: #’s 10-85, multiples of 5