trinomials factoring

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.


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.


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 FF and example #5 & #6 emphasize that the last term in the product is the result of LL, 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


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


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 –


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:

This pair of problems

emphasizes the

importance of checking

multiplication of signs

of the last terms of the

factors. See ‘check’

below each problem.

factors of 6: 1, 6 and 3, 2


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.


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: .


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


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.


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


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:


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 :


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:


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:


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:

D. Looking Ahead (with desmos.com) 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).


SSoouurrcceess::

desmos.com

mathtothecore.com

math.tutorvista.com

Using standard form, find the y-intercept:

Express in factored form and find the x-intercepts:

trinomials factoring

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