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Factoring

Factoring. GPS Algebra Unit 4. Essential Question. What methods can I use to factor quadratic equations? What is factor? What is factor form?. GPS Standard. MM1A3. Students will solve simple equations.

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Factoring

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  1. Factoring GPS Algebra Unit 4

  2. Essential Question What methods can I use to factor quadratic equations? What is factor? What is factor form?

  3. GPS Standard • MM1A3. Students will solve simple equations. • Solve quadratic equations in the form ax2+ bx+ c = 0, where a = 1, by using factorization and finding square roots where applicable. Later… Today

  4. Day 1 Objective • Factor second degree polynomials (quadratic functions) using guess and check and “the big X”

  5. Recall • Multiplying binomials: (x+1)(x+2) = Use FOIL! +2 x2 +1x +2x = x2+3x+2

  6. Factoring • Multiplying binomials in reverse • Starting with a quadratic function, your goal is to find the two binomials that will multiply to give you back your original quadratic • In a nutshell: Finding two binomial factors that will give you the product of the original quadratic equation

  7. Guess and Check • Guess what binomials you think might work, then multiply them together to see if you get the right answer • In a nutshell: Starting from factor form and using trial and error to “guess and check” factors “What is factor form Ms. Ross?” factor form factor “What is a factor Ms. Ross?”

  8. Definitions  • Factor – one of two numbers multiplied together to give you a product • Factor Form – writing factors in equation form • Solution - the answer to a problem 4 and 3 are factors of 12 Ex. 4 x 3 = 12 Ex. x2 + 7x + 12 = (x + 3)(x + 4)  Factor Form (x + factor)(x + factor) If we solve for x we would get: x = -3, -4  The Solution

  9. Method 1: Guess and Check Recall: Start from factor form and using trial and error to “guess and check” factors • Example 1: x2 + 9x -36 Factors of 36: Factor Form = (x + )(x + ) factor factor Which factors add to give you 9? (x + )(x + ) 3 - 12 3,12 6, 6 Answer:(x – 3)(x + 12) 4,9 2,18 1, 36

  10. Method 2: The BIG X add b X • ax2+ bx+ c factor factor c multiply =(x + factor)(x + factor)

  11. The BIG X may also look like this: add b X • ax2+ bx+ c u v c multiply =(x+u)(x+v)

  12. Practice add X • x2+ 5x + 4 5 1 4 4 multiply =(x+4)(x+1)

  13. Practice add X • x2+ 3x -10 3 5 -2 -10 multiply =(x+5)(x-2)

  14. Note: Whenever "b" is negative, you make the larger factor number negative in the BIG X. *If they are the same, both are negative* Practice -6 add X • X2 – 6x -16 -8 2 -16 multiply =(x+2)(x-8)

  15. Essential Question What methods can I use to factor quadratic equations? What is factor? What is factor form? Guess and Check Big X one of two numbers multiplied together to give you a product writing factors in equation form =(x + factor)(x + factor)

  16. Next Class… Day 2 Objectives: • Factor second degree polynomials using GCF (greatestcommonfactor)and grouping

  17. Factoring GPS Algebra Day 2 for Factoring

  18. Let’s Recall… GPS Standard • MM1A3. Students will solve simple equations. • Solve quadratic equations in the form ax2+ bx+ c = 0, where a = 1, by using factorization and finding square roots where applicable. Today Still 

  19. Essential Question What new methods can I use to factor quadratic equations? What is a GCF? What is grouping?

  20. Factoring Methods: • Guess and Check: Starting from factor form and using trial and error to “guess and check” factors • Big X: Today: GCF and Grouping add b X factor factor c multiply “What is a GCFMs. Ross?” “What is a Grouping Ms. Ross?”

  21. New Definitions  • GCF– Greatest Common Factor,there are factors common to each term we can factor them out of each term • Grouping– a form of backwards distribution 2x is the GCF = 2x(2x + 3) Ex. (4x2+6x) b(a + c) Ex. ab + cb =  Grouping We are going to do this with binomials

  22. To be sure that you have factored correctly do the multiplication 2x(3x-1)and see that you get the original expression back again. GCF Start by looking for common factors in each of its terms. If there are factors common to each term we can factor them out of each term. Example 1: xis also a common factor Here each term has 2 as a common factor we factor 2x This expression cannot be factored any further. Check: 2x(3x – 1) = 6x2 – 2x

  23. GCF Practice Simplify the following. 1.) 12x3 – 9x2 2.) 4x4 + 16x 3.) 3a2b4 + 9ab6 x2 = 3 ( ) x 4 - 3 x = 4 ( ) 1 + 4 x3 b4 a = 3 ( ) a b2 + 3

  24. Grouping Example Example 1: 6x3 + 2x2 + 12x + 4 Step 1: Group Binominals (6x3+ 2x2)+ (12x + 4) Step 2: Find the GCF of each binomial Step 3: Group binomials (12x + 4) (6x3 + 2x2) 2(3x3+ x2 ) 2(3x3 + x2 4(3x + 1)  This binomial is done Now this binomial is  done 2x2(3x + 1) Recall:ab + cb = b(a + c) Note that they end up with the same binomial 2x2(3x + 1) + 4(3x + 1) (3x + 1)(2x2+ 4) a b + c b b (a + c)

  25. Grouping Practice Group the following equations 1. 8r3 – 64 + r – 8 2. 12x3 + 2x2 – 30x – 5 3. 25v3 + 5v2 + 30v + 6

  26. Put it All together Methods of Factoring 1 – Guess and Check 2 – Big X 3 – GCF 4 – Grouping

  27. Essential Question What new methods can I use to factor quadratic equations? What is a GCF? What is grouping? GCF and Grouping Greatest Common Factor, there are factors common to each term we can factor them out of each term a form of backwards distribution b(a + c) Ex. ab + cb =

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