1. Factor trinomials of the form ax² + bx + c.

2. Factor out the GCF before factoring a trinomial of the form ax² +bx + c.

Factoring trinomials of the form ax² + bx + c (a ¹ 0) requires further consideration when a ¹ 1.  Remember to remove the GCF first.

Recall the FOIL method for multiplication of binomials.

1. Find two First terms with a product of ax².

                                ( ¦x + ) ( ¦x + ) = ax² + bx + c

                    2.   Find two Last terms with a product of c.

                       ( + ¦ ) ( + ¦ ) = ax² + bx + c

          3.  Repeat steps (2) and (3) until a combination is found for which the sum of the

                Outer and Inner products is bx.

                                ( ¦x + ¦ ) ( ¦x + ¦ ) = ax² + bx + c

Remember: If c is positive, both binomials will have the same sign.

                                ( + ) ( + ) or ( – ) ( – )

If c is negative, the binomials will have different signs.

                                ( + ) ( – ) or ( – ) ( + )

Example 1: 2x² + 11x + 12

( + ) ( + )         Signs are both positive since c is positive and bx is positive.

( 2x +  ) ( x +  )             2x · x = 2x²

( 2x + 3) ( x + 4)             3 · 4 = 12

8x+ 3x  = 11x                       Check the sum of the outer and inner products.

Therefore, ( 2x + 3) (x + 4) is the factored form of 2x² + 11x + 12.

Example 2:        2x² – 5x – 12

( + ) ( – )         Signs are different since c is negative.

(2x +? ) ( x -? )            2x · x = 2x²

( 2x + 3) ( x – 4)           3 · –4 = –12

-8x+ 3x= 5x                   Factors and signs must be arranged so that the sum of the outer and inner products is –5x.

Therefore, ( 2x + 3) (x – 4) is the factored form of 2x² – 5x – 12

Some trinomials which are not in the form ax² + bx + c may be factorable. Consider the example below.

(Use the coefficient of y² as you would c and the coefficient of xy as you would b.)

Trial and Error

3x² – 10xy + 3y²

(  –   ) (  –    )                Signs are both negative since c is positive and bx is negative.

(3x – ) (x – )                                 3x · x = 3x²

(3x – y) (x – 3y)                     (–y) · (–3y) = +3y²

– 9xy -  xy   = -10xy                     Check the sum of the outer and inner products.

Therefore, ( 3x – y ) ( x – 3y ) is the factored form of 3x² – 10xy + 3y².

Exercise Set 4.3

see homework assignment sheet for specific problems.