 # Factoring Decision Tree

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Factoring Decision Tree

Factoring Decision Tree
Expression Factoring Decision Tree GCF

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)

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

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

Using FOIL we find the product of two binomials.

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

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

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

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

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

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

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

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

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

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.

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.

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.

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

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.

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

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

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

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

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