Instead of simply memorizing rules & formulas, it really2
to understand why (for what purpose), you are adding,
subtracting, factoring, etc.
Remember, terms are separated by + or – signs. (The terms are
all being added or subtracted). An example is: 10x2
Factoring is finding an equal expression that is a product, that
is, the factors are multiplied. How do we factor 10x2
+ 15x ???
The first step is to always look for a common factor. Is there one
for this polynomial?We factor out the GCF of 5x. What remains
is (2x + 3). The full factor is 5x(2x + 3).
+ 15x = 5x(2x + 3)
We multiply factors to find the terms of a polynomial.
(These terms are either being added or subtracted.)
The Main Idea
Steps to Factoring Completely:
1. Look for the GCF
2. Look for special cases.
a. difference of two
b. perfect square
3. If a trinomial is not a perfect
square, look for two different
4. If a polynomial has 4 terms,
you can try to factor by grouping.
6. Not all polynomials can be
5. Understand what you are
working with before you attempt
A) Always be on the lookout for them
B) Be able to recognize them in both factored and
9x2 - 16x2
Factor these difference of squares:
18y3 – 8y
4m2 – 49n2 = (2m)2 – (7n)2 Difference of
(2m + 7n)(2m – 7m)
FOIL With a Positive and a Negative
(x + 3)(x - 5)
F= (x•x) = x2
O= (x•-5) -5x
I= (3•x) +3x
L= (3•5) 15
Answer: x2 - 2x - 15
The larger number is negative,
so the middle term is negative.
(x - 3)(x + 5)
FOIL With a Negative and a Positive
The opposite of our first example. The
larger number is positive, resulting in a
positive middle term.
Answer: x2 + 2x - 15
...But I thought if the signs were different, the middle
term was cancelled out?
This is true, but only if the two numbers are the same!
(3x + 5)(3x – 5)
To solve these, we use FOIL in reverse.
x2 + x – 6 = (x ) ( x )
Check the signs:
Then, list all the factors of the last term, looking for the
sum of the middle term, and the product of the last term.
Factoring Trinomials: (1x2 + bx + c)
(The leading coefficient is always 1)
x2 + x – 6 =
x2 + x – 6 =6
(x + )(x - )
(x + 3)(x - 2 )
No algebra magic or wizardry here. Factor
and check for the correct fit.
Factor the polynomial x2 + 13x + 30. That was too easy,
one moreFactor the polynomial x2 – 2x – 35.
Since our two numbers must have a product of – 35 and a sum of
– 2, the two numbers will have to have different signs.
Factors of – 35 Sum of Factors
– 1, 35 34
1, – 35 – 34
– 5, 7 2
5, – 7 – 2
Absolutely the last practice problem.
2ab2 – 26ab – 60a