P.4 FACTORING (التحليل) Objectives: Greatest Common Factor Factoring Trinomial Special Factoring Factoring by Grouping General Factoring

Like prime factorization of a number Only integer coefficients and constants allowed

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 factors factor out the GCF = 6t2

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 squares Cannot be factored with real coefficients “prime”

Perfect square – perfect square

b. Sum/Difference of Two Cubes: Ex. GCF

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 if is a perfect square.

Ex: 25a2 – 90ac + 81c2 Check 25a2 is a perfect square. 25a2 = 5a  5a 81c2 is a perfect square. 81c2 = (-9c)  (-9c) 2(5a)(-9c) = -90ac This is a perfect square trinomial. 25a2 – 90ac + 81c2 = (5a – 9c)2

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 = x2 We 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)

X2 + 8x + 12 The “+” 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 8 The “+” sign before the 12 also lets us know both signs in the solution will be the same.

X2 + 8x + 12 Possibilities for solution: (x - 6)(x - 2) or… (-6)(-2) = +12 and (6)(2) = +12 But -6 + -2 = -8; 6 + 2 = +8 Sooooo…. (x + 6)(x + 2) … check by FOIL

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!

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; 4 + 3 = 7 (-4)(-3) = +12; -4 + -3 = -7 … these are the factors we are looking for! (x - 4)(x - 3) Check by FOIL

X2 - 4x - 12 Leading coefficient is 1; we need two factors of 12 … whose difference is 4 The “-” 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 & 2 The “-” before the 4 lets us know the sign of the larger number (6) must be negative … (x - 6)(x + 2)

2x2 + 9x + 4 Perhaps 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.

3. Factoring by Grouping Factor mx Factor 2 Group the first two terms and the last two terms watch Factor mx Factor 2

Ex: Factor

You’re shining!