4.1 FACTORING POLYNOMIALS
Previously, you have multiplied algebraic expressions using algebra tiles and rectangular area
models. For example, you may have used an area model to write the product (x + 8)(x + 5) as
the sum x2 + 13x + 40.
Now you will focus on reversing this process: How can you write a product when given a sum?
The process of changing a sum to a product is called factoring. When an expression is written
as a product, it is said to be in factored form, and each of the expressions being multiplied is
called a factor. Can every expression be factored? That is, can you rewrite every sum as a
product?
I.
Common Term: When asked to factor completely, always look for a GCF (greatest
common factor) first.
Factor each of the following expressions.
a. 4 x 2  18
b. 15a 2  12a  9
c.
II.
25 x 3  30 x 2  5 x
Trinomials of the form ax 2  bx  c : One method used to factor such trinomials is the
diamond (sum & product) and area model approach. Once you have enough practice, you
may no longer need to use the diamond and area model when factoring – just make sure
you verify that the product of your factors equals the area as a sum. Following is a
diagram that begins this approach. Once you complete the inside of the area model, look
for common factors that could represent the dimensions.
a. n 2  9n  8
b. x 2  10 x  16
4.1 FACTORING POLYNOMIALS
c. x 2  3 x  28
e. 3 p 2  11 p  10
III.
d. 2 x 2  13x  6
f. 12 x 2  7 x  12
“Putting it all together”: Completely factor each of the following (don’t forget to look
for a GCF first!)
a. 5b 2  30b  80
b. 25 x 2  5 x  20
c. 6 x 2  48 x  42
d. 28n 2  44n  24
4.1 FACTORING POLYNOMIALS
e. 10  23 p  5 p 2
2
f. x y  10 y  3xy
IV.
Reminder: Your factors are the dimensions of an area model and their product should
always equal the sum of the parts inside the area model. Write the area of the rectangle
below as a sum and as a product of factors.
V.
Special “Products”: Use the diamond-area model approach to factor each of the
following.
a.
x2  9
b. x 2  8 x  16
4.1 FACTORING POLYNOMIALS
c. x 2  10 x  25
d. 4 x 2  25
e. 9r 2  12r  4
2
f. k  36
g. x 2  9
2
h. 25 x  4
Complete the table below for the special products you just factored.
a
d
f
g
h
b
c
e
Sum
x2  9
4 x 2  25
k 2  36
x2  9
25 x 2  4
x 2  8 x  16
x 2  10 x  25
9r 2  12r  4
Factors
Special Name
4.1 FACTORING POLYNOMIALS
VI.
Factor the following completely (think – GCF, then special product, then diamond-area
model):
b. 6 x 2  x  15
c. 4 x 2  81
d. 7 x 2  847
3
2
e. 5 x  15 x  20 x
f. x 2  16 x  64
g. 4 x 2  30 x  36
h. 2 x 2  9 x  9
i. 2 x 2  32
a.
x 2  x  42
j. 3x 2 y 2  xy2 4 y 2
2
2
k. 2 x  3xy  y