factoring decision tree. expression gcf step 1 gcf 14x + 21 = 9x – 12y = 2x 2 + 6x + 4 = 5ab 2 +...
TRANSCRIPT
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
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
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