115-0302-Factoring-173.pdf

115-0302-Factoring-173.pdf - Factoring You may have studied...

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Factoring You may have studied several methods of factoring earlier. Although there is nothing wrong with these, I want to stress the single most important method of factoring, namely, Guess and Check . Example 1 Factor: 6 x 2 ° 5 x ° 4 Solution) Guess 6 x 2 ° 5 x ° 4 = (? x + ?) (? x + ?) . To distinguish ? , put 6 x 2 ° 5 x ° 4 = ( ax + b ) ( cx + d ) Start from guessing a; c with ac = 6 , and b; d with bd = ° 4 . For example, guess a = 1 ; c = 6 ; b = 2 ; d = ° 2 . Then FOIL to check your solution: ( x + 2) (6 x ° 2) = 6 x 2 ° 2 x + 12 x ° 4 = 6 x 2 + 10 x ° 4 Unfortunately the middle term is 10 x , which is not equal to ° 5 x . You must do trial and error until you get the right one. Eventually you will °nd 6 x 2 ° 5 x ° 4 = (2 x + 1) (3 x ° 4) Of course 6 x 2 ° 5 x ° 4 = (3 x ° 4) (2 x + 1) is also correct. If all terms have a common factor, you should take it out °rst. Example 2 Factor: 3 x 2 + 6 x ° 72 Solution) 3 x 2 + 6 x ° 72 = 3 ° x 2 + 2 x ° 24 ± (Take out 3 °rst) = 3 ( x ° 4) ( x + 6) Note: It is much harder to factor directly as 3 x 2 + 6 x ° 72 = (? x + ?) (? x + ?) because 72 is big and there are many possibilities.
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