Factoring, The Fun Never Ends Rule 1: Take out all common factors – ALWAYS DO THIS FIRST (The Distributive Property) 3x2y – 6xy2 What’s common? ______ 3xy 3x2y – 6xy2 = 3xy (x – 2y) 2x4 – 8x2 What’s common? ______ 2x2 2x4 – 8x2 = 2x2 (x2 – 4) 5x + 10x2 – 40x3 What’s common? ______ 5x 5x + 10x2 – 40x3 = 5x (1 + 2x – 8x2) Rule 2: For TWO TERMS ( factor using the rules below or it does not factor) *if quadratic use difference of squares: x2 – y2 = (x – y)(x + y) 4 – a2b2 = ( ) ( ) 2 – ab 2 + ab 16c2 – 25a2 = ( ) ( ) 4c – 5a 4c + 5a 9x2 – y2 = ( ) ( ) 3x – y 3x + y 1 – 36x2y2 = ( ) ( ) 1 – 6xy 1+ 6xy *if cubic use difference of cubes: x3 – y3 = (x – y)(x2 + xy + y2) or sum of cubes: x3 + y3 = (x + y)(x2 – xy + y2) 8 – a3 = ( ) ( ) 2 – a 4 + 2a + a2 x3 + 8a3b3 = ( ) ( ) x + 2ab x2 – 2abx + 4a2b2 27x3 – 1 = ( ) ( ) 3x – 1 9x2 + 3x + 1 8y3 + 125 = ( ) ( ) 2y + 5 4y2 – 10y + 25 Notice: the trinomial terms come from the binomial terms, first term = binomial first term squared, middle term = product of both with change of sign, and last term = last squared. Rule 3: For THREE TERMS, Perfect Square Trinomials and General Strategy x2 + 2xy + y2 = (x + y)(x + y) = (x + y)2 x2 – 2xy + y2 = (x – y) (x – y) = (x – y)2 *Perfect Square Trinomial formula: 16 – 8x + x2 = (4 – x)(4 – x) = (4 – x)2 4x2 + 12xy + 9y2 = (2x + 3y)(2x + 3y) = (2x+ 3y)2

Factoring Rules continued Rule 3: For THREE TERMS – *General Strategy – explained using the following example Example: 4x2 – 7x – 15 Step 1- multiply outside numbers 4 and 15 getting 60 4·15 2·30 12·5 20·3 6·10 1· 60 Step 2- form all the two number products that equal 60 Step 3- select the product pair whose sum or difference is equal to the middle term coefficient in the given problem. (sum if 4 and 15 have the same sign, difference otherwise) difference is 7 Step 4- write the number pair as terms with x and with signs so that adding them gives the middle term, –7x : -12x + 5x = -7x Step 5- replace the –7x in the original with the two terms that equal it and group in pairs 4x2 – 12x + 5x – 15 = (4x2 – 12x) + (5x – 15) when possible put the negative term first Step 6- factor each pair using the distributive property: 4x (x – 3) + 5 (x – 3) now use the distributive once more getting the factors: (x – 3) (4x + 5) So factoring the trinomial 4x2 – 7x – 15 = (x – 3) (4x + 5) Rule 4: For FOUR TERMS – Group in pairs (steps 5 & 6 above) or group 3 terms & one as below 9 – a2 + 2ab – b2 = 9 – (a2 – 2ab + b2) notice how a parenthesis after a negative sign changes signs = 9 – (a – b)2 and using difference of squares = [3 – (a – b)][3 + (a – b)] So 9 – a2 + 2ab – b2 = (3 – a + b) (3 + a – b)

More examples, first we’ll finish the last two examples given with Rule 1: * 2x4 – 8x2 = 2x2 (x2 – 4) from Rule 1, but since the last factor is two terms and fits the difference of squares (Rule 2) the completely factored form is: 2x4 – 8x2 = 2x2 (x – 2) (x + 2) * 5x + 10x2 – 40x3 = 5x (1 + 2x – 8x2) from Rule 1, but since the last factor has three terms Rule 3 may be applied: 1 times 8 is 8, factors of 8 that have a difference of 2 are 4 and 2, -2x + 4x = 2x so 1 + 2x – 8x2 = 1 – 2x + 4x – 8x2 = (1 – 2x) + (4x – 8x2) = 1(1 – 2x) + 4x(1 – 2x), therefore 5x + 10x2 – 40x3 = 5x (1 – 2x) (1 + 4x) Notice the above two examples required more than one rule to factor completely, the procedure for factoring then is FIRST apply Rule 1, SECOND count the terms and use the correct rule to finish. There are some polynomials that cannot be factored more, for example all the trinomials in the examples for the sum or difference of cubes formulas, i.e. 9x2 + 3x + 1. Can you tell why? * 8x + 27x4 = x (8 + 27x3) = x (2 + 3x) (4 – 6x + 9x2) What rules were used? _________ Rules 1 & 2 sum of cubes * 2ax + 6a – bx – 3b = (2ax + 6a) – (bx + 3b) What rule is being used ______ Rule 4 Why is the negative in front of 3 changed to +? _____________________ Parenthesis inserted after minus sign = 2a (x + 3) – b (x + 3) = (x + 3) (2a – b) what property is used twice? _______ distributive * 25x2 + 70x + 49 = (5x + 7) (5x + 7) what rule? _____, the answer can also be written? ______ Rule 3 (5x + 7)2 * a6 – b6 = (a3 + b3) (a3 – b3) what rule? ______, which other rules can be applied? ________ Rule 2 Both cubic rules So a6 – b6 = (a + b) (a2 – ab + b2) (a – b) (a2 + ab + b2)