Math 209 – Week 2
Quick Reference Card
This Quick Reference Card (QRC) contains information about the concepts presented in Math 209 Week 2 (Chapter 6.1-6.4 of the
course textbook). For more information about any of these concepts, visit the Center for Math Excellence (CME). Even more
information about Math 209 Week 2 concepts can be found on the Web at sites such as www.JoleneMorris.com and
www.purplemath.com. As soon as your Math 209 class begins, you will also have access to FREE tutoring 24-hours a day at the
CME. Ask for tutoring in any of the Week 2 concepts that are not clear to you.
FACTORING POLYNOMIALS
Many years ago when I was teaching remedial mathematics, most of my students had trouble
remembering basic number facts such as the times tables (the multiplication facts). Calculators
are fine for doing large multiplication and division problems, but a person still needs to know the
basic number facts in order to understand our number system. Because my remedial math
students were 12 to 15-years-old, they had spent the last five to seven years trying to memorize
the times tables but still didn't know them. I needed to find a different way to teach the same
concepts so I used a worksheet that I called the "reverse times tables." I would give the answer
to a multiplication fact and ask students to give me the problem (see the sample to the right).
64 = _____ x ______
28 = _____ x ______
45 = _____ x ______
The types of problems you are going to be doing in college algebra this week are similar to those reverse times tables. You will be
given a polynomial and asked to factor it (find the expressions that were multiplied together to result in that polynomial). This "reverse
times tables" with polynomials is called factoring. Because factored polynomials are not basic facts that you can memorize (like the
multiplication facts), you need to develop skills to undo the multiplication that resulted in the polynomial. In essence, when you are
asked to factor a polynomial, you should understand that the polynomial is a result of a multiplication of other polynomials, and you
must find those original polynomials. There are several ways to factor a polynomial. Trial and error is a perfectly valid method;
however, trial and error could take hours to factor just one polynomial. Other methods we use are covered in this QRC.
(1) COMMON FACTORS
When factoring a polynomial, we first look for factors that are
common to each term. Look at both factors of the coefficient
and the variables.
EXAMPLE:
The coefficients of 8, 2, and 4 have a 2 in common
The variables of w3, w2, and w have a w in common
The greatest common factor of all terms is 2w
SOLUTION:
(2) FOUR TERMS  GROUPING
After removing any common factors, check to see if there are four
terms. Four terms are factored by grouping: Pair the first two terms
and pair the last two terms – add the two pairs. Remove any
common factors in each pair of terms. What is left should be a
common binomial so factor it out:
EXAMPLE:
First pair has 5x2 in common; second pair has –1 in common
Now both terms have (3x – 2) in common
SOLUTION:
(3) DIFFERENCE OF SQUARES
(4) SUM/DIFFERENCE OF CUBES
After removing any common factors, if there are two terms,
check to see if that binomial is a difference of squares:
After removing any common factors, if there are two terms, check
to see if that binomial is a sum or difference of cubes:
DIFFERENCE OF SQUARES  a2 – b2 = (a + b)(a – b)
SUM OF CUBES  a3 + b3 = (a + b)(a2 – ab + b2)
EXAMPLE:
DIFFERENCE OF CUBES  a3 - b3 = (a - b)(a2 + ab + b2)
First term is 22 times x2; second term is 32
Thus, this binomial is (2x)2 – (3)2
With a = 2x and b = 3
SOLUTION:
EXAMPLE:
First term is 23x3; second term is 53y3
Thus, this binomial is (2x)3 – (5y)3
With a = 2x and b = 5y
SOLUTION:
This Quick Reference Card prepared by Jolene M. Morris ([email protected])
(5) PERFECT SQUARE TRINOMIALS
(6) UN-FOIL (TRIAL & ERROR)
After removing any common factors, if there are three terms,
check to see if that trinomial is a perfect square. If the first and
third terms are squares, check the second sign, which must be
positive, and check the middle term, which must be 2ab:
After removing any common factors, if there are three terms
and that trinomial is not a perfect square, you must “un-FOIL”
the trinomial by trial and error. Be sure to view the factoring
tips mini-movie on the JoleneMorris.com website for tips
on factoring by trial and error. Trial and error works best
when the coefficient of the first term is one.
PERFECT SQUARE TRINOMIAL 
PERFECT SQUARE TRINOMIAL 
EXAMPLE:
EXAMPLE:
The factors of 14 are 1 × 14 and 2 × 7
1 + 14 ≠ 9 but 2 + 7 = 9 (the middle term’s coefficient)
a = 2x; b = 3; middle term = 2ab = 12x
(2x)2 – 2(2x)(3) + (3)2
SOLUTION:
SOLUTION:
(8) POSAMENTIER’S ALGORITHM
(7) ac METHOD
The ac method of factoring is trial & error but with educated
guessing. The ac method works best when the coefficient of
the first term is greater than one. Find the product of a and c.
List all factor pairs of ac that add to the coefficient of the
middle term (b). Re-write the trinomial as four terms, and then
factor using grouping.
EXAMPLE:
ac = 10; The factors of 10 are 1 × 10 and 2 × 5
2 + 5 ≠ 11 but 1 + 10 = 11 (the middle terms), so the
middle term can be written as 1x + 10x:
Factor by grouping:
)
Another method similar to trial & error and the ac method is
Posamentier’s algorithm. This algorithm is seldom taught but
may help in factoring complicated trinomials.
1.
2. Rewrite as
where yz = ac and y + z = b
3. Replace y and z in the fractional form from #2 above
4. Factor both binomials, where possible.
5. Simplify to 1 with the denominator. NOTE: if the denominator
doesn’t cancel completely, the trinomial is PRIME
=
EXAMPLE:
yz = -24; y + z = +5; thus, y = +8 and z = -3
SOLUTION:
=
NOTE: If a trinomial can’t be factored, it is PRIME.
SOLUTION:
SUMMARY OF FACTORING
1. Factor out the greatest common factor of all terms
2. If the polynomial has four terms, try factoring by grouping
3. If the polynomial is a binomial, look for difference of squares
4. If the polynomial is a binomial, look for difference of cubes
5. If the polynomial is a binomial, look for sum of cubes
6. If the polynomial is a trinomial, check for a perfect square
7. Otherwise apply FOIL in reverse using trial and error if a = 1
8. Otherwise use the ac method if a > 1
9. Alternately, use Posamentier’s Algorithm. Refer to
Posamentier, A.S. and Stepelman, J. (2002). Teaching
secondary mathematics: Techniques and enrichment units (6th
Edition). Columbus, Ohio: Prentice-Hall for more details on this
algorithm.
SQUARES & CUBES
PRIME FACTORIZATIONS