Factoring Polynomials: ax2 + bx + c
Trial and Error
Box Method
1. Set up parenthesis ( __ + __ )( __ + __ )
1. Draw a box:
divided into four parts
2. Make two lists. The first is a list of the sets of
factors for the first term of the polynomial. The 2. Fill in the top left hand corner with the leading
second is a list of the sets of factors for the last
term. Fill in the bottom right hand corner with
term in the polynomial.
the last term.
9x2
5
3x2
2
2
3x
+
17x
–
6
9x + 14x + 5
x, 9x
1,5
–6
3x, 3x
3. Multiply the leading coefficient with the last
3. Fill in the first blank in each parenthesis with a
term coefficient. Then find the set of factors that
set of factors from the first list. Fill in the
add up to the middle coefficient.
second blank in each parenthesis with a set of
Factors
Sum
factor from the second list (3x + 1)(3x + 5)
3(-6) = -18
9, -2
7
18,
-1
17
4. Check your factorization by multiplying to see
if you get the original polynomial.
2
(3x + 1)(3x + 5) = 9x + 8x + 5  Wrong
This is not our original polynomial so we have to
try new factors. Keep trying new combinations of
factors until you get your original polynomial.
(9x + 5)(x + 1) = 9x2 + 14x + 5  Right
This is our original polynomial so we are done.
Example:
6x2
6x2 – 7x + 2
x, 6x
2x, 3x
2
1, 2
4. Fill in the other two boxes. These two boxes
must add up to your middle term so attach your
variable. (No, order does not matter.)
3x2 18x
3x2 + 17x – 6
-1x – 6
5. Factor out the common factor from each row 
and column . Take the sign of the box right
next to the common factor.
Your answer is the binomial
x
+3
across the top times the
3x 3x2 18x
binomial down the side.
-1 -1x – 6
3x2 + 17x – 6 =(x + 3)(3x – 1)
Example:
2
(6x – 2)(x – 1) = 6x – 8x + 2  no
(3x – 1)(2x – 2) = 6x2 – 8x + 2  no
2
8x + 18x + 9
8(9) = 72 = 6(12)
6 + 12 = 18
(3x – 2)(2x – 1) = 6x2 – 7x + 2  yes
4x
+3
2x + 3
8x2 12x
6x
9
8x2 + 18x + 9 = (2x + 3)(4x + 3)
Directions:
Factor.
1} 4y2 + 7y – 2 =________________________
2} 4x2 – 17x – 15 =______________________
3} 6x2 – 7x + 2 =________________________
4} 75k2 – 30k + 3 =______________________
5} 3v2 – 8v – 3 =________________________
6} 8m2 + 2m – 3 =_______________________
7} 6 + 19x + 15x2 =______________________
8} 3p2 – 16pq – 12q2 =___________________
9} 2 + 8x + 6x2 =________________________
10}6n2 – 23np + 7p2 =____________________