Methods of Factoring
General Strategy for Factoring (Guess and Check)
Factor 20 x  3x 2  7
1) Arrange the polynomial in descending order.
2) If the leading coefficient is negative, or a greatest
common factor exists then factor this out from the
entire polynomial (but don’t lose it!)
3) To factor a trinomial of the form ax 2  bx  c , you
want to write it as a product of two binomials:
ax 2  bx  c  (__ x  ____)(__ x  ____)
Try combinations of the factors of a (in this case 3)
in the first term of each binomial and combinations
of c (in this case -7) in the second term of each
binomial. If c is positive, the signs within the
binomial factors match the sign of b; if c is negative,
the signs within the binomial factors are opposite.
4) Check by multiplying.
 3x 2  20 x  7
 1(3x 2  20 x  7)
 1(1x  1)(3x  7)   1(3x 2  20 x  7)
 1(1x  1)(3x  7)   1(3x 2  20 x  7)
 1(3x  1)( x  7)   1(3x 2  20 x  7)
 1(3x  1)( x  7)   1(3x 2  20 x  7)
So the answer is
 1(3x  1)( x  7)
FOIL to get  1(3x 2  20 x  7)
Distribute the -1 to get  3x 2  20 x  7
Factoring by Grouping
Factor 44  42 x  2 x 2
1) Arrange the polynomial in descending order.
 2 x 2  42 x  44
2) If the leading coefficient is negative, or a greatest
common factor exists then factor this out from the
entire polynomial (but don’t lose it!)
3) To factor a trinomial of the form ax 2  bx  c , find
two numbers (p and q) whose product is a*c (in this
case -22) and whose sum is b (in this case -21). (if c
is positive, the numbers have the same sign; if c is
negative, the number have different signs)
5) Then rewrite the trinomial breaking the middle term
into px + qx.
x 2  bx  c  ax 2  px  qx  c .
 2( x 2  21x  22)
6) Then look at the first two terms [ ax 2  px ] and
factor out of them both their greatest common
factor.
7) Repeat with the second two terms qx  c . If the
trinomial is factorable then the binomials in
brackets should be the same [in this case (x – 22)]
8) Then factor this binomial out to get your final
answer.
9) Check by multiplying.
p * q = -22
p + q = -21
p = -22
q =1
 2( x 2  21x  22) =  2( x 2  22 x  1x  22)
 2( x( x  22)  1x  22)
 2( x( x  22)  1( x  22))
 2( x  22)( x  1)
FOIL to get  2( x 2  21x  22)
Distribute the -2 to get  2 x 2  42 x  44
Table Method for Factoring
3a2 + ab – 10 b2
* Put the first term with its sign diagonally
under the Factor and the third term with its
sign diagonally under that, in the lower right
hand corner.
Step 1
Factor
* Get the diagonal product by multiplying the
terms that are now in the table.
Step 2
Diagonal Product = (3a2)( – 10 b2) = –30a2 b2
* Write factors of the coefficient of the
diagonal product. (Note: There are two sets
Step 3
–1•30
–2•15
–3•10
–5•6
because the product is negative.)
* Find the factors that add to the middle
term (in this case +1)
* Write these factors, with the variable(s)
of the middle term of the original problem, in
the lower left and upper right cells.
Step 4
Factor
* Put the greatest common factor (gcf) of
each row in the cell to the left of the row.
Step 5
3a2
– 10 b2
1•(–30)
2•(–15)
3•(–10)
5•(–6)
3a2 -5ab
6ab – 10 b2
Factor
gcf of 3a and -5ab is a a
3a2 -5ab
gcf of 6ab and – 10 b2is
2b
6ab – 10 b2
2b 
2
* Put the greatest common factor (gcf) of
each column in the cell above the column.
* You can now read the factors from the
outside of the table:
(a +2b) is in the left column
(3a-5b) is in the top row
Although this method is practically foolproof,
you should always, always check by
multiplying.
gcf of 3a2
and 6ab is
3a
↓
Factor 3a
a
3a2
gcf of 3a2
and -5ab is
a
gcf of 6ab
2b
6ab
and – 10
b2is
2b 
Step 6
Check:
(a +2b) (3a-5b) = 3a2 + ab – 10 b2
gcf of -5ab
and – 10 b2 is
-5b
↓
-5b
-5ab
– 10 b2