Math 1400 - Manyo
Review 5: Factoring Polynomials - page 1 of 2
Review 5 – Factoring Polynomials
Type 1:
, where and are integers
We want to factor
into two factors of the form
where and
positive or negative. If we complete the multiplication, we get (and you should verify this)
Therefore,
can be either
and
The first clue in factoring this type of expression is to determine if is positive or negative.


If is positive, then either both and are positive or both and are negative.
o The sign of and is determined by the sign of . If is negative, and are negative.
If is positive, and are positive. And, is the sum of and (i.e.
)
If is negative, then and have opposite signs.
o The sign of is the same as the sign of
.
Example 1:
Factor
into
In this example
and
. We need two factors of whose sum is -5. Since
is positive we
know that and are either both negative or both positive. Since
is negative, we know that and
are both negative. Again, we need two factors of whose sum is -5. The possible factors are shown in the
chart to the left:
Since the last entry in the table satisfies
6
1
2
3
-6
-1
-2
-3
Example 2:
Factor
o
o
We conclude
into
(Notice the similarity to Example 1)
In this example
and
. Since
is negative we know that and have opposite signs.
Since
is negative, the factor with the larger magnitude is negative.
We need two factors of
whose sum is -5. The possible factors are shown in the chart to the left:
Since the third entry in the table satisfies
6
-1
2
-3
-6
1
-2
3
o
o
We conclude
Math 71 - Manyo
Review 5 - page 2 of 2
Type 2:
: the difference of two squares
The difference of two squares always factors into the product of two conjugates
Definition: conjugates are two binomials such that one is the sum of two terms and the other is the difference
of the two terms.
Example 3:
Factor
Example 4:
Factor
Type 3: any polynomial expression
Look for common factors in all terms of the polynomial, which include common constant factors and variable
factors .
Example :
o
o
o
Factor
The three terms have constant factors of 6, 4 and 10, which have a common factor of 2
The three terms have variable factors of
, which have a common factor of
Therefore, we can pull a factor or
from each of the three terms in the expression.