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Section The Greatest Common Factor and Factoring by Grouping

Section The Greatest Common Factor and Factoring by Grouping. The answer to a multiplication problem is called a product . Numbers that will multiply together to give a product are called factors of the product. List all of the factors of 36. 1  36 = 36 2  18 = 36 3  12 = 36

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Section The Greatest Common Factor and Factoring by Grouping

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  1. SectionThe Greatest Common Factor andFactoring by Grouping

  2. The answer to a multiplication problem is called a product. Numbers that will multiply together to give a product are called factors of the product. List all of the factors of 36. 1  36 = 36 2  18 = 36 3  12 = 36 4  9 = 36 6  6 = 36 Factors 1, 2, 3, 4, 6, 9, 12, 18, 36

  3. List the Factors of 30 1  30 = 30 2  15 = 30 3  10 = 30 5  6 = 30 1, 2, 3, 5, 6, 10, 15, 30

  4. Compare the factors of 30 and 36. 1, 2, 3, 5, 6, 10, 15, 30 Factors of 30 1, 2, 3, 4, 6, 9, 12, 18, 36 Factors of 36 What are the common factors of 30 and 36? Which is the greatest? 1, 2, 3, 6 6 is the greatest common factor of 30 and 36 6

  5. Common Factor • An integer that is a factor of two or more integers is a common factor of those integers. • 6 is a common factor of 30 and 36. • 1, 2, and 3 are also common factors of 30 and 36. • 6 is the greatest common factor of 30 and 36.

  6. Write each number in factored form. Example: Find GCF of 36 and 48 36 = 12 · 3 48 = 12 · 4 GCF =12 Finding the Greatest Common Factor for Numbers

  7. Find the GCF of 30, 20, 15 Since 5 is the only common factor it is also the greatest common factor GCF. 30 = 6 · 5 20 = 4 · 5 15 = 3 · 5

  8. Find the GCF of 10, 21, 15 Since 1 is the only common factor it is also the greatest common factor GCF. 10 = 1 · 2 · 5 21 = 1 · 3 · 7 15 = 1 · 3 · 5 Usually 1 is not included in the list of factors but it was needed this time to show a common factor.

  9. Find the GCF of 6m4, 9m2, 12m5 6m4 = 2 · 3 · m2 · m2 9m2 = 3 · 3 · m2 12m5 = 4 · 3 · m2 · m3 GCF = 3m2

  10. Find the GCF of 5x and 15 GCF = 5 5x = 5 · x Distribution is the way to check factoring. 15 = 3 · 5 5x + 15 = 5( + ) Factored Form = 5(x + 3) = 5x + 15

  11. Factor out the greatest common factor. First write down the GCF Divide each coefficient by 2 2y2 (5y3 - 4y2 + 3) 10y5 - 8y4 + 6y2 = Decrease each exponent by 2 Keep the same signs

  12. Factor out the greatest common factor. 2y2 (5y3 - 4y2 + 3)= 10y5 - 8y4 + 6y2 Check with distribution

  13. Factor out the greatest common factor. 8p5q2 + 16p6q3 - 12p4q7 = 4p4q2 ( ) 2p + 4p2q - 3q5

  14. Factor Out the Common Factor ax + bx = x ( a + b ) (x + 5) ( a + b ) a(x + 5) + b(x + 5) =

  15. Factor Out the Common Factor ( x + 7 ) x(3a + 2) + 7(3a + 2) = (3a + 2)

  16. Factor By Grouping 1. Group First Two Terms and Group the last Two Terms. 2. Find Greatest Common Factor of First Two Terms 3. Find Greatest Common Factor of Second Two Terms 4. Write Down What Is Common to Both 5. Write Down What Is Left Over

  17. Factor by Grouping 6y2 - 8yw + 15yw - 20w2 = (6y2 - 8yw) + (15yw - 20w2) = 2y(3y - 4w) + 5w(3y - 4w) = (3y - 4w)(2y + 5w) =

  18. Factor 6x2 – 3x – 4x + 2 by grouping 6x2 – 3x – 4x + 2 = (6x2 – 3x) + (– 4x + 2) = 3x(2x – 1) + -2(2x - 1) = (2x – 1)(3x – 2)

  19. Your Turn Factor xy + 2x + 4y + 8 by grouping xy + 2x + 4y + 8 = (xy + 2x) + (4y + 8) = x(y + 2) + 4(y + 2) = (y + 2)(x + 4)

  20. Try Again Factor 6x2 – 9x – 4x + 6 by grouping 6x2 – 9x – 4x + 6 = (6x2 – 9x) + (– 4x + 6) = 3x(2x – 3) + -2(2x – 3) = (2x – 3)(3x – 2)

  21. What Did You Learn? • After factoring a polynomial how can you check your result? • When would you try to factor by grouping?

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