P.4 FACTORING (التحليل) Objectives: Greatest Common Factor
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1 P.4 FACTORING (التحليل) Objectives: Greatest Common Factor Factoring TrinomialSpecial FactoringFactoring by GroupingGeneral Factoring
2 Like prime factorization of a number Only integer coefficients and constants allowed
3 Methods of Factoring: Ex: Factor out the GCF. Def: Factoring means to write a polynomial as a product of polynomials of lower degree.Methods of Factoring:1. Factor Out the Greatest Common Factor( GCF)GCF = product of all prime factors raised to the smallest powers.Ex: Factor out the GCF.write each term as product of its prime factorsfactor out the GCF = 6t2
4 2. Try to Factor A binomial by One of the Following Special Factoring Formulas: a. Difference of Two Squares:Ex.No similar rule for a sum of squaresCannot be factored with real coefficients “prime”
7 a. As a perfect –square trinomial b. Using the trial method Factorization Theorem:The trinomial with integer coefficients a, b and c can be factored as the product of two binomial with integer coefficients if and only ifis a perfect square.
8 Ex: 25a2 – 90ac + 81c2Check25a2 is a perfect square. 25a2 = 5a 5a81c2 is a perfect square. 81c2 = (-9c) (-9c)2(5a)(-9c) = -90acThis is a perfect square trinomial.25a2 – 90ac + 81c2 = (5a – 9c)2
9 Ex: Check Try Trail and Error Method Smile! The leading coefficient is 1! Easy!Set up for FOIL( )( )We know First term has to be x because only x*x = x2We know last term has to be factors of 12 … 12,1; or 4,3; or 6,2.(x 12)(x 1) or (x 4)(x 3) or (x 6)(x 2)
10 X2 + 8x + 12The “+” sign before the “12” lets us know we will be adding the two factors.The sum of the 2 factors must = +8!Of the 3 pairs of factors only 6 and 2 have a sum of 8The “+” sign before the 12 also lets us know both signs in the solution will be the same.
12 X2 - 7x + 12 Here again our leading coefficient is 1… (x )(x ) The last terms must be factors of 12 … 6,2; or 12,1; or 3,4.The “+” before the 12 tells us we will be adding the 2 factors, and that the signs will be the same!The sum of the factors must be -7!
13 X2 - 7x + 12 Of the 3 pairs of factors of 12, only 4 & 3 sum to 7 Signs must be the same, so…(4)(3) = +12; = 7(-4)(-3) = +12; = -7 … these are the factors we are looking for!(x - 4)(x - 3)Check by FOIL
14 X2 - 4x - 12Leading coefficient is 1; we need two factors of 12 … whose difference is 4The “-” sign in front of the 12 also tells us that the signs will be different in our solution.Factors of 12 whose difference is 4 … 6 & 2The “-” before the 4 lets us know the sign of the larger number (6) must be negative …(x - 6)(x + 2)
15 2x2 + 9x + 4Perhaps by now we can recognize that both signs in the factors will be “+”In this case we only have 3 possibilities(2x + 4)(x + 1)(2x + 1)(x + 4)(2x + 2)(x + 2)Check by FOIL to see which is a solution.
16 3. Factoring by Grouping Factor mx Factor 2 Group the first two terms and the last two termswatchFactor mxFactor 2