1. Factoring Decision Tree

2. Factoring Decision Tree
Expression
Factoring Decision Tree
GCF

3. Step 1 GCF 14x + 21 = 9x – 12y = 2x2 + 6x + 4 = 7(2x + 3) 3(3x – 4y)
Expression
Step 1 GCF
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)

4. Factoring Decision Tree
Expression
Factoring Decision Tree
GCF
Count the number of terms
Two Terms Binomial
Difference of Squares

5. Difference of Squares a2 – b2 = (a + b)(a – b) x2 – 9 = (x + 3)(x – 3)
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

6. Using FOIL we find the product of two binomials.

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

8. Factoring Decision Tree
Expression
Factoring Decision Tree
GCF
Count the number of terms
Two Terms Binomial
Difference of Squares
Three Terms Trinomial
Special Pattern

9. Special Patterns
Using FOIL we find the product of two binomials.

10. Rewrite the perfect square trinomial as a binomial squared.
So when you recognize this…
…you can write this.

11. Recognizing a Perfect Square Trinomial
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

12. Recognizing a Perfect Square Trinomial
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

13. Recognizing a Perfect Square Trinomial
Signs must match!
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

14. Recognizing a Perfect Square Trinomial
Not 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

15. Factoring Decision Tree
Expression
Factoring Decision Tree
GCF
Count the number of terms
Two Terms Binomial
Not a Special Pattern
Three Term Trinomial
Difference of Squares
Leading Coefficient = 1
Special Pattern
Grouping

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

17. Factoring a Trinomial by Grouping
First list the factors of 24.
Rewrite with four terms.
Now add the factors.
1 24 25
2 12 14
3 8 11 10
4 6
Notice that 4 and 6 sum to the middle term.

18. Factoring a Trinomial by Grouping
First list the factors of 24.
Rewrite with four terms.
Now add the factors.
1 24 25
2 12 14
3 8 11 10
4 6
Notice that 2 and 12 sum to the middle term.

19. Factoring Decision Tree
Expression
Factoring Decision Tree
GCF
Count the number of terms
Two Terms Binomial
Not a Special Pattern
Three Term Trinomial
Difference of Squares
Leading Coefficient = 1
Leading Coefficient ≠ 1
Special Pattern
Grouping

20. Coefficient a ≠ 1
First list the factors of 2∙(-38) = -76.
Rewrite with four terms.
Now subtract the factors.
1 76 75
2 38 36
4 19 15
Notice that 4 and 19 do the job.

21. Factoring Decision Tree
Expression
Factoring Decision Tree
GCF
Count the number of terms
Two Terms Binomial
Not a Special Pattern
Three Term Trinomial
Difference of Squares
Leading Coefficient = 1
Leading Coefficient ≠ 1
Special Pattern
Grouping
Inspection

22. Inspection Guess at the factorization until you get it right.
Check with multiplication.
With practice this is the quickest.

23. Factoring Decision Tree
Expression
Factoring Decision Tree
Four Terms
GCF
Grouping
Count the number of terms
Two Terms Binomial
Not a Special Pattern
Three Term Trinomial
Difference of Squares
Leading Coefficient = 1
Leading Coefficient ≠ 1
Special Pattern
Grouping
Inspection

24. Four Term Grouping
If the polynomial has more than three terms, try to factor by grouping.

25. Factoring Decision Tree
Expression
Factoring Decision Tree
Four Terms
GCF
Grouping
Count the number of terms
Two Terms Binomial
Not a Special Pattern
Three Term Trinomial
Difference of Squares
Leading Coefficient = 1
Leading Coefficient ≠ 1
Special Pattern
Grouping
Inspection