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Factoring Lesson: GCF, Square Trinomials, Difference of Squares

This lesson covers factoring using the greatest common factor (GCF), identifying and factoring square trinomials, and identifying and factoring difference of two squares. The lesson includes examples and practice problems.

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Factoring Lesson: GCF, Square Trinomials, Difference of Squares

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  1. Lesson 5-4 & 5-5: Factoring Objectives: Students will: Factor using GCF Identify & factor square trinomials Identify & factor difference of two squares

  2. Day 1 • Trinomials

  3. If there is a GCF factor it out!!!!!! Ex: 2x + 8 2(x + 4) Trinomial (3 terms) Polynomial (4 terms) Binomial (2 terms) Is it a Perfect Square Trinomial? A2 ± 2AB + B2 ex: 4x2-20x +25 (2x-5)2 Is it a difference of squares or cubes ? A2- B2 or A3±B3 ex: 4x2 – 25 or x3 - 64 yes No No PST A2 +2AB+B2 = (A + B)2 Or A2 -2AB+B2 = (A - B)2 Ex: 4x2 –20x +25 = (2x - 5)2 Find Done If a=1 If a≠1 Factor by: Grouping Or Undo foil ( )( ) or box Write out factors Rewrite as four terms Difference of Squares (DS) A2- B2 = (A +B )(A – B) Ex: 4x2 – 25 = (2x + 5)(2x - 5) Difference (or sum) of Cubes (A3 – B3) = (A - B)(A2 +AB + B2) Or (A3 + B3) = (A + B)(A2 - AB + B2) (then factor trinomial if possible) Ex: x3 – 64 = (x – 4)(x2 + 4x + 16) Repeat with (ax-b) if possible factor flow chart Remember: Number of exponent tells you number of Factors/ Solutions/ Roots/ Intercepts x1 = 1 factor x2 = 2 factors x3 = 3 factors x4 = 4 factors and so on…..

  4. Factoring The reverse of multiplying 2x(x+3) = 2x2 + 6x So: 2x2 + 6x = Look for GCF of all terms → numbers & variables ► Reverse distribute it out → DIVISION Example 1 Factor 6u2v3 – 21uv2 What is the GCF? Pull out GCF (divide both terms) 3uv2 3uv2(2uv - 7)

  5. Factoring 4-term • Make Sure Polynomial is in descending order!!!!!!!! • 3 Methods • Reverse FOIL • F O I L • x2 + 5x + 4x + 20 • ( )( ) • REMEMBER: ALWAYS FACTOR A GCF 1st IF YOU CAN Find GCF of first two terms- fill first spot Find what makes up ( F) and fill in first spot in other factor already have x so need another x Move to outside (O) already have x so need + 5 Move to inside (I) already have x so need + 4 Check last (L) 4x5 =20 so done!! x + 4 x + 5

  6. x2 + 5x + 4x + 20 • Foil Box ( x + 5)(x + 4) x + 5 x + 4

  7. B) Factor by grouping Find GCF of first two terms- and factor out Find GCF of second two terms- and factor out What is in parenthesis should match –so factor it out Write what is left as other factor x2 + 5x + 4x + 20 x( x + 5 ) + 4(x + 5) (x + 5) (x - 4) It’s the same either method!! I like the FOIL method. What do you think????

  8. ax2 + bx + c – A General Trinomial Where does middle term come from? (x + 2)(x + 3) = x2 + 3x + 2x + 6 (2x + 4)(x – 3) = 2x2 - 6x + 4x – 12 2x2 - 2x - 12 So to factor we are unFOILing!!

  9. Steps for General Trinomial Factoring 1) Factor out GCF (always first step) 2) Find product ac that add to b table (to find O and I) 3) Write middle term as combo of factors ( 4 terms) 4)Unfoil or by grouping Example 1: x2 + 7x + 12 F O I L ( )( ) 12 1) no GCF x2 + 4x + 3x + 12 2) x + 3 x + 4 1*12 13 2*6 8 3*4 7

  10. TRY Example 2 Factor x2 – 5x – 24 Example 3 Factor x2 – 12x + 27

  11. EX 4) Harder One -24 6x2 – 5x – 4 -8x 6x2 + 3x - 4 F O I L ( ) ( ) 3x - 4 2x + 1 GCF of first 2

  12. Factor: -7a + 6a2 -10

  13. Factor: 56 + x – x2

  14. Assignment (day 1) • 5-5/227/ 22-72 e

  15. Day 2

  16. Factoring Perfect Squares, Difference of Square, Look back at the forms for each of these from Lesson 5-3 Factor the following: Ex 1: x2 – 8x + 16 Perfect Square Trinomial so Ex 2: 9x2 – 16y2 Difference of squares so (x - 4)2 (3x + 4y)(3x – 4y)

  17. Don’t forget GCF! Ex 3: Factor 8x2 – 8y2

  18. x2 – 2xy + y2– 25 (x-y)2 - 25 ((x-y) + 5)((x-y)– 5) Trick: Ex 4: Combo perfect square trinomial and difference of squares Apply PST Now apply DS

  19. Ex 5: Factor:

  20. Marker Board pg 222-223 • 21 • 33 • 41 • 51

  21. ASSIGNMENT • 5-4/222-223/18-62e, 86-92 e

  22. Day 3 • Sum or Difference of cubes

  23. Review Cubing Binomials • (a+b)3= (a+b)(a2 +2ab+b2) a3 +3a2b+3ab2+b3 (similarly for (a-b)3)

  24. Notice all the middle terms cancelled out like DS. What were the terms that cancelled? What are the factors? Example 1: (a3 + b3) a2 -ab + b2 a3 -a2b ab2 a +b a2b -ab2 b3 (a3 + b3)= ( a+b)(a2-ab+b2) Is the remaining trinomial factorable?

  25. Ex 2: Factor 27x3-8y3 or (3x)3 _ (2y)3 +4y2 + 6xy 9x2 3x -2y 27x3 18x2y +12xy2 -8y3 -18x2y -12xy2 27x3-8y3=(3x-2y)(9x2+6xy+4y2) A3 – B3 = (A-B)(A2 + AB+ B2)

  26. Ex 3: Factor x3 + 64

  27. Formulas A3 – B3 = (A-B)(A2 + AB+ B2) A3 + B3 = (A+B)(A2 - AB+ B2)

  28. Factor : 125x3 +1

  29. Marker Board pg 227 • 1 • 13 • 19

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