Factoring Decision Tree

Factoring Decision Tree

Expression

GCF

Step 1 GCF

14x + 21 =

9x – 12y =

2x2 + 6x + 4 =

5ab2 + 10a2b2 + 15a2b =

7(2x + 3)

3(3x – 4y)

2(x2 + 3x + 2)

5ab(b + 2ab + 3a)

Expression

GCF

Factoring Decision Tree

Expression

GCF

Count the number of

termsTwo Terms Binomial

Difference of Squares

2 2 ( )( )a b a b a b

Difference of Squares

• If the polynomial has two terms (it is a binomial), then see if it is the difference of two squares.

a2 – b2 = (a + b)(a – b)x2 – 9 = (x + 3)(x – 3)

• Remember the sum of squares will not factor in the real numbers. a2 + b2

Using FOIL we find the product of two binomials.

)5)(5( xx

2x x5 x5 25252 x

Rewrite the polynomial as the product of a sum and a difference.

)5)(5( xx252 x

Factoring Decision Tree

Expression

GCF

Two Terms Binomial

Difference of Squares

2 2 ( )( )a b a b a b

Count the number of

terms

Three Terms Trinomial

Special Pattern

Special Patterns

))(( baba

2a ab ab 2b22 2 baba

Using FOIL we find the product of two binomials.

Rewrite the perfect square trinomial as a binomial squared.

))(( baba 22 2 baba 2)( ba

So when you recognize this…

…you can write this.

Recognizing a Perfect Square Trinomial

25102 xx• First term must be a perfect square.

(x)(x) = x2

• Last term must be a perfect square.

(5)(5) = 25• Middle term must be twice the product of the roots of the first and last term.

(2)(5)(x) = 10x

2)5( x

Recognizing a Perfect Square Trinomial

1682 mm• First term must be a perfect square.

(m)(m) = m2

• Last term must be a perfect square.

(4)(4) = 16• Middle term must be twice the product of the

roots of the first and last term.

(2)(4)(m) = 8m

2)4( m

Recognizing a Perfect Square Trinomial

81182 pp• First term must be a perfect square.

(p)(p) = p2

• Last term must be a perfect square.

(9)(9) = 81• Middle term must be twice the product of

the roots of the first and last term.

(2)(-9)(p) = -18p

2)9( p

Signs must match!

253036 2 pp

Recognizing a Perfect Square Trinomial

• First term must be a perfect square.

(6p)(6p) = 36p2

• Last term must be a perfect square.

(5)(5) = 25• Middle term must be twice the product of

the roots of the first and last term.

(2)(5)(6p) = 60p ≠ 30p

Not a perfect square trinomial.

Factoring Decision Tree

Expression

GCF

Two Terms Binomial

Difference of Squares

2 2 ( )( )a b a b a b

Count the number of

terms

Special Pattern

2)( ba 22 2 baba

Leading Coefficient = 1

Not a Special PatternThree

Term Trinomial

Grouping

mnxmnx )(2mnmxnxx 2

24102 xx

Grouping: Start with the trinomial and pretend that you have a factorization.

))(( nxmx

This means that to find the correct factorization we must find two numbers m and n with a sum of 10 and a product of 24.

24102 xx

Factoring a Trinomial by Grouping

First list the factors of 24.

1 242 12

3 8

4 6

Now add the factors.

25

14

11

10

Notice that 4 and 6 sum to the middle term.

)6)(4( xx

2x x6 x4 24Rewrite with four terms.

2( 6 ) (4 24)x x x ( 6) 4( 6)x x x

( 6)( 4)x x

24142 xx

Factoring a Trinomial by Grouping

First list the factors of 24.

1 242 12

3 8

4 6

Now add the factors.

25

14

11

10

Notice that 2 and 12 sum to the middle term.

)12)(2( xx

2x x2 x12 24)2412()2( 2 xxx

)2(12)2( xxx)12)(2( xx

Rewrite with four terms.

Factoring Decision Tree

Expression

GCF

Two Terms Binomial

Difference of Squares

2 2 ( )( )a b a b a b

Count the number of

terms

Special Pattern

2)( ba 22 2 baba

Leading Coefficient = 1

Not a Special PatternThree

Term Trinomial

Grouping

Leading Coefficient ≠ 1

22 15 38x x Coefficient a ≠ 1

First list the factors of 2∙(-38) = -76.

1 762 38

4 19

Now subtract the factors.

75

36

15

Notice that 4 and 19 do the job.

(2 19)( 2)x x

22x 19x 4x 38

Rewrite with four terms.

2(2 19 ) ( 4 38)x x x (2 19) 2(2 19)x x x

(2 19)( 2)x x

Factoring Decision Tree

Expression

GCF

Two Terms Binomial

Difference of Squares

2 2 ( )( )a b a b a b

Count the number of

terms

Special Pattern

2)( ba 22 2 baba

Leading Coefficient = 1

Not a Special PatternThree

Term Trinomial

Grouping

Leading Coefficient ≠ 1

Inspection

Inspection•Guess at the factorization

until you get it right.•Check with multiplication.•With practice this is the

quickest.

Factoring Decision Tree

Expression

GCF

Two Terms Binomial

Difference of Squares

2 2 ( )( )a b a b a b

Count the number of

terms

Special Pattern

2)( ba 22 2 baba

Leading Coefficient = 1

Not a Special PatternThree

Term Trinomial

Grouping

Leading Coefficient ≠ 1

Inspection

Four Terms

Grouping

Four Term Grouping

• If the polynomial has more than three terms, try to factor by grouping.

axaxx 5522 2

)55()22( 2 axaxx )(5)(2 axaxx

)52)(( xax

Factoring Decision Tree

Expression

GCF

Two Terms Binomial

Difference of Squares

2 2 ( )( )a b a b a b

Count the number of

terms

Special Pattern

2)( ba 22 2 baba

Leading Coefficient = 1

Not a Special PatternThree

Term Trinomial

Grouping

Leading Coefficient ≠ 1

Inspection

Four Terms

Grouping