The document provides instruction on factoring trinomials. It reviews factoring by GCF and difference of squares. It then explains that trinomial factoring involves finding two binomial factors whose product is the trinomial. When the leading coefficient is 1, the factors of the last term must combine to give the middle term. When the leading coefficient is greater than 1, all combinations of the last term's factors must be tried. Special cases like a GCF, multiple patterns, or non-factorable trinomials are also addressed. Examples demonstrate the process and applications include finding dimensions and areas.

1. Marianne McFadden
April/May 2015
Factoring Trinomials
REVIEW OF FIRST TWO FACTORING PATTERNS:
We have learned about two methods of factoring so far.
First, we discovered that factoring out the GCF (GREATEST COMMON FACTOR) is the reverse of the distributive property and some examples
we studied include:
 Greatest Common Factor (GCF)
Next, we investigated factoring the DIFFERENCE OF TWO SQUARES into the product of two binomials, as reviewed below:
 Difference of Two Squares
                    
The next factoring pattern will also produce two binomials as factors, but the original problem to be factored will be a trinomial, not a
binomial (as with the difference of two squares). Some of the problems we will learn to factor will look like:
TRINOMIAL FACTORING
 Trinomial factoring involves polynomials which have three terms, arranged in
standard form (from highest to lowest exponent of a single variable). If the trinomial
has a GCF, it is removed before using the factoring pattern (example #6 & #8). Exponents
in the first and last term of the original problem are found in the middle term, but are
halved (like , or A study of the pattern and its factors follows.

 
 
 
Recall that GCF factoring creates a distributive property problem
where the term to be distributed (the GCF) is the most that can be
divided out of the original polynomial. The polynomial that
remains (after the GCF has been removed) should have NO other
common factors.
Difference of Two Squares factoring involves two terms that are perfect squares: the
coefficients and constants must be perfect squares (1, 4, 9, 16, ...) and the exponents
must be even. The factors created in this pattern are identical except for the middle
signs, and because these signs differ, when checking by FOIL multiplication, the
Outer and Inner terms are opposites and cancel, leaving just the difference between
the products of the First and Last terms.

2. Marianne McFadden
April/May 2015
In order to figure out how to factor a trinomial, it would be beneficial to review multiplying binomials so we can look for patterns in
helping us to reverse the process. Take a look at the following samples which involve the four-step multiplication process performed in
multiplying binomials (also referred to as FOIL, meaning FIRST, OUTER, INNER, LAST):
math.tutorvista.com
mathtothecore.com
Note: The 4-step process produces
only three terms when simplified. It
is easy to see that:
 first  first = the 1st
term of the
trinomial and
 last  last = the 3rd
term of the
trinomial, but the middle term of the
trinomial is a combination of the
OUTER and INNER products. Study the
examples that follow to discover a
relationship that will aid in factoring.

3. Marianne McFadden
April/May 2015
 F O I L  F O I L  F O I L
 F O I L  F O I L  F O I L
  
ALL examples above are expanded FOIL problems that we have completed in the unit on multiplying polynomials. Examples #1 & #2 emphasize that the first term in
the product is the result of FF and example #5 & #6 emphasize that the last term in the product is the result of LL, and each of these four examples have matching
signs so the last term of the product is positive while the sign of the middle term of the product matches the signs of the “L” terms in the binomials.
Examples #3 & #4 emphasize that the outers and inners combine to result in the middle term of the trinomial product. Since the signs are opposites, the middle term is
obtained by subtracting the “O” and “I” and takes the sign of the higher (absolute value) term. Now we can focus on factoring or “un-FOILing trinomials.
A. Trinomial Factoring ⇛ (
LET’S START with factoring trinomials that have a leading coefficient of 1. This is the easier type of trinomial since the relationship between
the middle term’s coefficient and the constant (or the coefficient of the last term) is easy to understand. Let’s look at example #1, #3, and
#5 from the above table and express each resulting trinomial as the product of its two binomial factors (orREVERSE THE FOIL):
trinomial factors We have to find the relationship between the second values in each binomial factor and the
constant term and the middle coefficient in the original trinomial.
SO, how are +4 and +3 related to +12 and +7?
How are and +5 related to and +1?
How are and related to and ?
Factors can be represented by ALGEBRA TILES:
becomes:
 We see that and .
Also, )( ) = and
Lastly, and SO...
 FACTORS OF THE LAST TERM MUST ALSO COMBINE TO
OBTAIN THE COEFFICIENT OF THE MIDDLE TERM 

4. Marianne McFadden
April/May 2015
So, we can think of pairs of values that have a specific product and sum or difference in helping us to factor trinomials.
Let’s practice: Two values, and are given. What two values have as their product AND as their sum or difference?
   Since and , the two values are .
CHECK YOUR ANSWERS: ; therefore,
; likewise,


Some students find it helpful to arrange the last term
(or coefficient of the last term) and the coefficient of the
middle term in a diagram as shown to the right, then
find the combination that multiplies to obtain the top
value AND combines to obtain the bottom value.
SO, in order to factor , we would first
place the 36 in the top section and the -15 in the bottom
section of the diagram, then find a pair of values that
multiply to obtain +36 AND combine to obtain -15. can be factored as: since
AND
+16 -2? ??
Problem #4 in the above problem set would
become: .
If the diagrams help you in getting the correct
pair of factors more quickly, then consider
using them.
-12 -3
, since
AND

5. Marianne McFadden
April/May 2015
Now we can try a few problems where the leading coefficient is 1.
Factor the following trinomials:
FACTOR: Note: ALL problems are
trinomials expressed in
standard form. Order of
the factors in the answer
does not matter. Watch
signs and placement of
variables!
 CHECK YOUR ANSWERS: 
OR
OR
                    
B. Trinomial Factoring ⇛ (
Now we can take trinomial factoring to the next level where THE LEADING COEFFICIENT IS GREATER THAN ONE. We will discover the
relationship between the arrangement of the factors of the last term (or its coefficient) and the combination of the Outer and Inner
products in FOIL. Study the comparison made below.
What is the difference between the following two trinomials?
AND
The first trinomial has a leading coefficient of 1, so it is easily factored – Find factors of +6 that will combine to equal +7? Our factors of 6
are +1 and +6, -1 and -6, +3 and +2, or -3 and -2. Since (+1)(+6) = 6 AND +1 + (+6) = 7, then
However, in the second trinomial, since , when we set up to factor the leading term, we get non-matching factors:
Next, we need to factor the last term (+6) and consider all the possibilities –

6. Marianne McFadden
April/May 2015
our goal is to obtain the middle term, , by combining the OOuutteerr and IInnnneerr products:
OUTER AND INNER
PRODUCTS
COMBINE ‘O’ AND ‘I’ TERMS:
(+)
Check (FOIL):
⇛
Can you find two of the four arrangements above that can be eliminated before considering the ‘O’ and ‘I’
products? Hint: Since original polynomial had NO GCF, the factors have no GCF also. See highlighted terms above.
Now we know that when the leading coefficient > 1, we need to try all differing combinations of the factors of the last term until we find
the combination that produces ‘outers’ and ‘inners’ (products) that will combine to equal the middle coefficient. Study the following
problems, noting any shortcuts suggested and the step-by-step method (sometimes referred to as trial and error) explained for each
example:
 Factor:

7. Marianne McFadden
April/May 2015
 Factor:
The next set of problems help to emphasize the importance of checking ‘sign multiplication’ as a final step in the process:
 Compare the factors of and Note that the original trinomials differ only by the sign of the third term.

8. Marianne McFadden
April/May 2015
 When the coefficient of the leading term is NOT prime, it is best to consider all combinations as F  F factors. So if the first term is ,
you should set up your factor options as:
 This next example is similar to example #3, but its factors are identical. Factor: .

9. Marianne McFadden
April/May 2015
 Note how many combinations are not considered in the next examples since some factor choices introduce a GCF that did NOT appear
in the original trinomial. Factor: and .
Keep in mind that this factor pattern takes time and practice to master. If you arrange the terms in standard form, set up all your options,
and try all possibilities, the process will become easier each time.
FACTOR THE FOLLOWING: CHECK YOUR ANSWERS:
   
   
   
   
B. Special Cases
GCF as a first step: Some trinomials have a GCF that must be factored first. Some samples:
 SSTTEEPPSS::
 STEPS:
factor the GCF factor the GCF
factor the trinomial factor the trinomial

10. Marianne McFadden
April/May 2015
More than one factoring pattern: Some trinomials have more than one factoring pattern, as seen in the examples below:
 SSTTEEPPSS::
 SSTTEEPPSS::
factor the trinomial factor the GCF
factor the difference of factor the trinomial
two squares factor the difference of two squares
Perfect trinomial squares: Sometimes factors match so the factors can be written as a square of a binomial:
 SSTTEEPPSS::
factor the trinomial – factors are identical
rewrite as a binomial squared
Prime (non-factorable) polynomials: Some problems cannot be factored – they are ‘prime’ (or non-factorable):
 try or ; cannot obtain from either combination
 try or eliminate – they will contain a GCF – obtain ???
Terms need to be rearranged: Make sure terms are arranged in standard form:
 SSTTEEPPSS::
 SSTTEEPPSS::
factor the GCF factor the GCF
arrange terms in standard form remove negative; put in standard form
factor the trinomial
factor the trinomial; rewrite as a binomial
squared
⇛ ⇛ ⇛ ⇛ ⇛ ⇛ ⇛ ⇚ ⇚ ⇚ ⇚ ⇚ ⇚ ⇚
Using the above examples as guides, factor the following polynomials.

11. Marianne McFadden
April/May 2015
Factor the following trinomials completely. Remember to factor the GCF first, if the problem contains one. Be on the lookout for
problems that need to be arranged in standard form and those that contain more than one type of factoring pattern. Also, there are a few
prime problems as well.
FFAACCTTOORR CCOOMMPPLLEETTEELLYY:: CCHHEECCKK YYOOUURR AANNSSWWEERRSS::
SET 1
 
 
 
 
 
 
 
 
 
 
SET 2
 
 
 
 
 
 
 
 
 
 

12. Marianne McFadden
April/May 2015
C. Applications of Factoring Trinomials
 The trinomial is factorable and is an integer
where > 0. List three possible values for , then write the three
resulting trinomials. Factor each trinomial.
workspace:
 The trinomial is factorable and has NO GCF. The
value of is an integer where < 0. List two possible values for
, then write the two resulting trinomials. Factor each trinomial.
workspace:
 A student factored Is the
trinomial factored correctly? Explain. If it is incorrect, then
factor correctly and check.
workspace:
 A student claimed that the trinomial Is this
possible? Explain. workspace:

The rectangle shown on the left has the indicated area. Find the dimensions of the rectangle by
factoring the trinomial completely. Express the dimensions as factors of the trinomial,
keeping in mind that the length is longer than the width. Describe how the length and width are
related to each other. workspace:

13. Marianne McFadden
April/May 2015
 Zach is considering redesigning his workroom. It is currently a square room as shown below. He would like it to be rectangular and
is thinking about making the width two feet shorter and making the length two feet longer than the original dimensions. He believes
that the workroom area will be the same since the amount taken from the width is the same as the amount added to the length. Is he
right - will the new rectangular space have the same area? Show your work – find both areas. If they are not equal, find the
difference between the areas and state which is larger.
workspace:

A woodworker wants to display his finished dining room furniture pieces in his store
showroom. The showroom has two identically sized adjoining rooms, each painted
similarly with two shades of a beige-colored neutral tone. Although the furniture on
display depicts two differing styles, the storeowner would like to accent the rooms with
a continuous wall border that is to be applied to the perimeter of the entire space (both
rooms attached or just the larger rectangle). workspace:
Find the following – area shown above is expressed in ft2 and :
 the dimensions of the rooms, in factored form, where the length is longer than the width
 the perimeter that will be covered, in terms of , and in factored form
 how many rolls of border need to be ordered to cover the perimeter, if each roll covers

(entire figure)
(inside area)
?
??
original
new
A landscaper is preparing a portion of a garden that is beside a historical landmark. The
entire shaded figure to the left represents the portion that will be worked on, and has an
area represented by The shaded portion inside (smaller, brownish
area) the larger rectangle has an area of . The smaller area will be dug
out and prepared to install a small fountain, surrounded by some low, ornamental plants
and the surrounding greenish area will be seeded for a grassy border.
Find the following, noting that :

14. Marianne McFadden
April/May 2015
workspace:
 Nathan has a large storage area in his basement and he wants to section off a portion of it so he can store just lawn tools and
ornaments in the smaller, enclosed area. Noting that the length of each shaded area shown is longer than the width:
(smaller area only)
(ENTIRE area, including smaller rectangle)
 the length and width of the entire area to be worked on; express in factored form
 the length and width of the area that is to be prepared for the fountain and plants
 the area of the border only (surrounding greenish area)
Compare the lengths and widths of both figures shown in order to find (numerical value):
 the width between the side of the smaller area and the side of the entire figure (the ‘?’)
 the length between the bottom of the smaller area and the bottom of the entire figure
(the ‘??’)
If each bag of grass seed covers , then compute:
 how many bags need to be purchased so that all the surrounding ground gets seeded;
explain your answer
 find the dimensions of the original storage area and the
dimensions of the enclosed area by factoring the area
expressions
 label the dimensions of each area on the figure
 compare the two width expressions and find the exact
width of the enclosed area ( ; explain your
computation
 using your computed value of above and its
corresponding expression in terms of , find the value of
.
 Find all the exact dimension values by evaluating the
expressions for the value of . Label the exact dimension
values on the figure; find exact area of the enclosed area
and the non-enclosed area.
Nathan will paint the floor of the non-enclosed area with an
anti-skid paint, and each can covers 80 – 120 ft2. He has one
can – will he be able to cover the area? If so, will he have
enough for the enclosed area too? Explain.
(areas shown are in ft2)
workspace:

15. Marianne McFadden
April/May 2015
 A room with a hardwood floor has an area represented by A decorative area rug is to be placed and centered on
the floor so that the border all around is uniform. The dimensions of the rug are represented by . Find the
exact width of the uniform border. Show all steps and explain.
total area of floor =
rug dimensions:
workspace:

16. Marianne McFadden
April/May 2015
D. Looking Ahead (with desmos.com)
 standard form
Expressed in factored form, it becomes:
Study the relationship between these
forms and the points named on the graph, the intercepts.
Using standard form, find the y-intercept:
Express in factored form and find the x-
intercepts:
Using standard form, find the y-
intercept:
Express in factored form and find the x-
intercepts:
Let’s take a look at some graphs of factorable trinomials. Now that we know how to
factor them, we can discover some relationships between specific points on the
graph and the forms in which the trinomials are expressed (standard or factored).

17. Marianne McFadden
April/May 2015
SSoouurrcceess::
desmos.com
mathtothecore.com
math.tutorvista.com
Using standard form, find the y-intercept:
Express in factored form and find the x-intercepts: