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**Factoring Decision Tree**

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**Factoring Decision Tree**

Expression Factoring Decision Tree GCF

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**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)

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**Factoring Decision Tree**

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

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**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

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

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

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**Factoring Decision Tree**

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

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

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**Rewrite the perfect square trinomial as a binomial squared.**

So when you recognize this… …you can write this.

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**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

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**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

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**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

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**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

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**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

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**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.

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**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.

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**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.

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**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

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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.

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**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

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**Inspection Guess at the factorization until you get it right.**

Check with multiplication. With practice this is the quickest.

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**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

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

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**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

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