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Factoring Completely: Learn to Factor Quadratic Expressions from a given GCMF

Learn how to factor completely and express as the product of prime factors, including finding the GCMF of monomials, using GCMFs to factor quadratic expressions, and dividing out the GCMF from polynomials.

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Factoring Completely: Learn to Factor Quadratic Expressions from a given GCMF

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  1. Lesson 10.4B : Factoring out GCMF Factor Completely – to express as the product of prime factors Ex. Factor completely : 24 6 4 2  2  2  3 2 3 2 2 Factor the following completely: 1) 5x2 2) 14x2y3

  2. GCF = 6 GCF of 12 and 18 GCF of 12x4y3 and 18xy5 GCF = 6xy3 GCMF : Greatest Common Monomial Factor – The greatest monomial that is a factor (will divide EVENLY into) of all the given monomials.

  3. To find the GCMF of two or more Monomials • First find the GCF of the coefficients • Find the largest power of each variable that is COMMON to all the monomials • The GCMF = product of GCF of coefficients and common variable factors

  4. Ex. Find GCMF of 12x2 and 18x GCF of coefficients = 6 Common variable(s) : only have one x in common GCMF = 6x Find GCMF of 21x2 and 35x5 GCF of Coefficients = 7 Common variable factors : two x’s GCMF = 7x2

  5. Find GCMF of 24x2y3 and 36x3y GCF of coefficients = 12 Common variable factors : two x’s and one y GCMF = 12x2y NOW, we are going to use GCMF’s to Factor Quadratic Expressions. Factoring Out the GCMF is the inverse (un-doing) of the Distributive Property

  6. To factor – undoing distributive property 1) Perform Distributive Property: 6(2 + 3) 12 + 18 Factor : 12 + 18 6 (2 + 3) 2) Use Distributive Property to simplify: 3(x + 7) 3x + 21 Factor: 3x + 21 3(x + 7) 3) Factor: 12x2y – 14xy3 2xy(6x – 7y2)

  7. Ex.Distribute 3x(x + 5) Means to multiply the 3x through the (x + 5) 3x(x) + 3x(5) 3x2 + 15x Ex. Factor 3x2 + 15x Means to Divide the GCMF out of the polynomial (divide each term by GCMF) GCMF = 3x Recall how to divide by monomial Divide (3x2 + 15x) by GCMF (3x) Factored form is 3x(x + 5)

  8. To factor a polynomial by factoring out the GCMF: • Find the GCMF • Divide the polynomial (each term of the polynomial) by the GCMF • Write the polynomial as the product of the GCMF and the result from step #2

  9. Example: Factor • 15x2 – 9 • Step 1) GCMF = 3 • Step 2) Divide 15x2 – 9 by the GCMF Step 3) Write as a product of GCMF and result of step 2 3(5x2 – 3)

  10. Factor • 28a3-12a2 • GCMF = 4a2 2) 15a – 25b + 20 GCMF = 5 Factored Form 5(3a-5b+4) Factored Form 4a2(7a – 3) 3) 16x5 – 14x3 + 26x2 GCMF = 2x2 Factored Form 2x2(8x3 – 7x + 13)

  11. Homework : Worksheet

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