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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 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….. factor flow chart If there is a GCF factor it out!!!!!! Ex: 2x + 8 2(x + 4) Trinomial (3 terms) Binomial (2 terms) Is it a difference of squares or cubes ? A2- B2 or A3±B3 ex: 4x2 – 25 or x3 - 64 yes Polynomial (4 terms) Is it a Perfect Square Trinomial? A2 ± 2AB + B2 ex: 4x2-20x +25 (2x-5)2 No No PST Find A2 +2AB+B2 = (A + B)2 Or A2 -2AB+B2 = (A - B)2 Ex: 4x2 –20x +25 = (2x 5)2 Done Difference of Squares (DS) = (A +B )(A – B) Ex: 4x2 – 25 = (2x + 5)(2x - 5) A2- B2 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) ac If a=1 Write out factors b If a≠1 Rewrite as four terms Factor by: Grouping Or Undo foil ( )( box ) or 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 Pull out GCF (divide both terms) 3uv2(2uv - 7) What is the GCF? 3uv2 Factoring 4-term Make Sure Polynomial is in descending order!!!!!!!! 3 Methods Find GCF of first two terms- fill first spot A) Reverse FOIL F O I L x2 + 5x + 4x + 20 ( x + 4 )( x + 5 ) 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!! REMEMBER: ALWAYS FACTOR A GCF 1st IF YOU CAN • Foil Box x2 + 5x + 4x + 20 ( x + 5)(x + 4) x +5 x F O x2 I +4 + 4x + 5x L + 20 B) Factor by grouping x2 + 5x + 4x + 20 x( x + 5 ) + 4(x + 5) (x + 5) (x - 4) 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 It’s the same either method!! I like the FOIL method. What do you think???? 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 12 Example 1: x2 + 7x + 12 1) no GCF 2) ac 1*12 b 7 1*12 13 2*6 8 3*4 7 x2 + 4x + 3x + 12 F O I L ( x + 3 )( x + 4 ) TRY Example 2 Factor Example 3 Factor x2 – 5x – 24 x2 – 12x + 27 EX 4) Harder One -24 6x2 – 5x – 4 6x2 -8x F O + 3x - 4 I ( 2x + 1 ) ( 3x GCF of first 2 L - 4 ) 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 (x - 4)2 Ex 2: 9x2 – 16y2 (3x + 4y)(3x – 4y) Difference of squares so 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: 64 ( x 2 8x 16) Marker Board pg 222-223 • • • • 21 33 41 51 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) Notice all the middle terms cancelled out like DS. What were the terms that cancelled? What are the factors? a2 a a3 +b a2b -ab -a2b -ab2 (a3 + b3)= ( a+b)(a2-ab+b2) + b2 ab2 b3 Is the remaining trinomial factorable? Ex 2: Factor 27x3-8y3 or (3x)3 _ (2y)3 9x2 + 6xy +4y2 +12xy2 3x 27x3 18x2y -2y -18x2y -12xy2 27x3-8y3=(3x-2y)(9x2+6xy+4y2) A3 – B3 = (A-B)(A2 + AB+ B2) 3 -8y Ex 3: Factor x3 + 64 Formulas 3 A – 3 B = 2 (A-B)(A + AB+ 2 B) A3 + B3 = (A+B)(A2 - AB+ B2) Factor : 125x3 +1 Marker Board pg 227 • 1 • 13 • 19