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

Day 1 Trinomials

factor flow chart If there is a GCF factor it out!!!!!! Ex: 2x + 8 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….. If there is a GCF factor it out!!!!!! Ex: 2x + 8 2(x + 4) Binomial (2 terms) Trinomial (3 terms) Polynomial (4 terms) Is it a difference of squares or cubes ? A2- B2 or A3±B3 ex: 4x2 – 25 or x3 - 64 Done No Difference of Squares (DS) A2- B2 = (A +B )(A – B) Ex: 4x2 – 25 = (2x + 5)(2x - 5) Repeat with (ax-b) if possible 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) Is it a Perfect Square Trinomial? A2 ± 2AB + B2 ex: 4x2-20x +25 (2x-5)2 PST A2 +2AB+B2 = (A + B)2 Or A2 -2AB+B2 = (A - B)2 Ex: 4x2 –20x +25 = (2x - 5)2 yes Find Write out factors If a=1 If a≠1 Rewrite as four terms Factor by: Grouping Or Undo foil ( )( ) or box ac b

The reverse of multiplying 2x(x+3) = 2x2 + 6x So: 2x2 + 6x = 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)

Make Sure Polynomial is in descending order!!!!!!!! 3 Methods 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

Foil Box + 5x + 20 + 4 x2 + 5x + 4x + 20 ( x + 5)(x + 4) x + 5 x F x2

It’s the same either method!! 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????

So to factor we are unFOILing!! 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!!

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) ac 1*12 b 7 x + 3 x + 4 1*12 13 2*6 8 3*4 7

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

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

Factor: -7a + 6a2 -10

Factor: 56 + x – x2

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

Day 2

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)

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

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

Ex 5: Factor:

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

Day 3 Sum or Difference of cubes

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

Example 1: (a3 + b3) (a3 + b3)= ( a+b)(a2-ab+b2) 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?

Ex 2: Factor 27x3-8y3 -8y3 A3 – B3 = (A-B)(A2 + AB+ B2) 27x3-8y3=(3x-2y)(9x2+6xy+4y2) A3 – B3 = (A-B)(A2 + AB+ B2)

Ex 3: Factor x3 + 64

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

Factor : 125x3 +1

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