Problem 2 Finding Common Factors

What is the expression in factored form?

1. 6 n 2 + 9 n 6 , bold italic n squared , plus 9 bold italic n

   6 n 2 + 9 n = 3 n ( 2 n ) + 3 n ( 3 ) Factor out the GCF , 3 n .   = 3 n ( 2 n + 3 ) Use the Distributive Property. table with 2 rows and 3 columns , row1 column 1 , 6 n squared , plus 9 n , column 2 equals 3 n open 2 n close plus 3 n open 3 close , column 3 cap factor out the cap gcap ccap f . comma 3 n . , row2 column 1 , , column 2 equals 3 n open 2 n plus 3 close , column 3 cap use the cap distributive cap property. , end table

2. 4 x 2 + 20 x − 56 4 , bold italic x squared , plus 20 bold italic x minus 56

   4 x 2 + 20 x − 56 = 4 ( x 2 ) + 4 ( 5 x ) − 4 ( 14 ) Factor out the GCF , 4 .   = 4 ( x 2 + 5 x − 14 ) Use the Distributive Property.   = 4 ( x − 2 ) ( x + 7 ) Factor the trinomial. table with 3 rows and 3 columns , row1 column 1 , 4 x squared , plus 20 x minus 56 , column 2 equals 4 open , x squared , close plus 4 open 5 x close minus 4 open 14 close , column 3 cap factor out the cap gcap ccap f . comma 4 . , row2 column 1 , , column 2 equals 4 open , x squared , plus 5 x minus 14 close , column 3 cap use the cap distributive cap property. , row3 column 1 , , column 2 equals 4 open x minus 2 close open x plus 7 close , column 3 cap factor the trinomial. , end table

✓ Got It?

2. What is the expression in factored form?
   1. 7 n 2 − 21 7 n squared , minus 21
   2. 9 x 2 + 9 x − 18 9 x squared , plus 9 x minus 18
   3. 4 x 2 + 8 x + 12 4 x squared , plus 8 x plus 12

Problem 3 Factoring a x 2 + b x + c eh , x squared , plus b x plus c  when | a | ≠ 1 vertical line eh vertical line not equal to 1

What is the expression in factored form?

1. 2 x 2 + 11 x + 12 2 , bold italic x squared , plus 11 bold italic x plus 12

   • Step 1 Since there is no common factor, find ac.

     ac = 2(12) = 24

   • Step 2 Since both b and ac are positive, find positive factors of 24 that have sum 11.

     Think

     How should you make your table in this case?

     Use the first row to list sets of factors of ac. Use the second row as before, to find the sum of each set of factors.

     Factors of 24	1, 24	2, 12	3, 8	4, 6
     Sum of factors	25	14	11	10

   • Step 3 Factor the trinomial as follows.

     2 x 2 + 11 x + 12 2 x 2 + 3 x + 8 x + 12 Rewrite  b x  using  b = 3 + 8 . x ( 2 x + 3 ) + 4 ( 2 x + 3 ) Find a common factor for the first two terms and another common factor for the last two terms. ( x + 4 ) ( 2 x + 3 ) Rewrite using the Distributive Property. table with 4 rows and 1 column , row1 column 1 , 2 x squared , plus 11 x plus 12 , row2 column 1 , 2 x squared , plus 3 x plus 8 x plus 12 , column 2 cap rewrite , b x , using , b equals 3 plus 8 . , row3 column 1 , x open 2 x plus 3 close plus 4 open 2 x plus 3 close , column 2 table with 2 rows and 1 column , row1 column 1 , cap find a common factor for the first two terms and , row2 column 1 , another common factor for the last two terms. , end table , row4 column 1 , open x plus 4 close open 2 x plus 3 close , column 2 cap rewrite using the cap distributive cap property. , end table