In my experience
teaching this class -- CHAPTER 4 IS THE KEY TO SUCCESS. Everything
we have done up to this point is about preparing us to work with
factoring polynomials (Chapter 4). Students that have developed
some comfort, confidence and accuracy with the previous 4 chapters (R-3)
will be able to complete this chapter.
In many way; Chapter 5, Rational Expressions revisits
Chapter 4 and extends student learning. It is rare to see a
student struggle with Chapters 5, 6, 7, 8, and 9 if they have mastered
Chapter 4. Please apply yourself fully to this unit -- in my
humble opinion, it is the key to Success in Beginning Algebra 41.
It will also be the key to success starting Algebra 141 with a solid
foundation of skills to build on.
4.1 The Greatest Common Factor
In many ways, factoring is the REVERSE of
multiplications. 2*3=6. Two factors of 5 are 2 and 3.
The process of writing a polynomial as a product is called factoring.
5x2+25x is a binomial (a polynomial with 2
terms. We can write this binomial as a product:
5x(x+5). This is an example of factoring.
Suppose we had started with the product 5x(x+5) -- when
we perform that multiplication (distributed property), we get 5x2+25x.
Do you see why factoring is the reverse of multiplication?
Definitions:
Factoring: process of writing an expression
as a product.
Greatest common factor (GCF) of a list of common
variables raised to powers is the variable raised to the smallest
exponent in the list (please refer to examples, page 271 in text).
Greatest common factor (GCF) of a list of terms
is the product of all common factors (please refer to examples, pages
272-273 in text).
Factor by Grouping:
-
Group terms into 2 groups of 2 terms each.
-
Factor out the GCF from each group.
-
If there is a common binomial factor, factor it out.
-
In not, rearrange the terms and try steps 1-3 again.
Please see examples pages 273-174 in text.
4.2 Factoring Trinomials in form x2+bx+c
In this section, we will look at trinomials that have a
"lead" coefficient of "1." Note that we do not
write coefficients of 1, they are assumed.
Factoring is writing as a project, remember that
multiplication is commutative -- we can factor the trinomial x2+7x+10
as either (x+2)(x+5) or (x+5)(x+2). To see why, multiply each of
those 2 factors (FOIL) and you will get the original expression when you
write the answer in descending order.
The key to factoring trinomials in the form x2+bx+c
is to remember that we will end up with a product of 2 binomials (x+?1)
and x+?2) where ?1 and ?2 are numbers
whose project is c. Please see examples, pages 280-281.
Note how the signs form a pattern. A positive
constant in a trinomial (c) means to look for 2 numbers (?1
and ?2) that have the same sign (positive and positive or
negative and negative). The sign of the coefficient of the middle
term tells us whether the signs are both positive or both negative.
A negative constant in a trinomial means to look for 2
numbers with opposite signs. Remember, that is the only way that
the result of our product can be negative. Please see examples,
page 282.
Factoring Out the Greatest Common Factor.
ALWAYS REMEMBER that the first step in factoring any
polynomial is to factor out the greatest common factor (if other than 1
or -1). DO NOT FORGET that you have pulled out a common factor
when this is possible. YOUR FINAL ANSWER MUST INCLUDE THIS GCF.
Please see examples, page 282.
Factoring Trinomials in the Form ax2+bx+c
Hopefully, starting with trinomials in the form x2+bx+c
allowed us to start looking at factoring patterns by considering the
simplest case -- "a" or the lead coefficient equaling 1.
Page 287 illustrate the process of rewriting ax2+bx+c
as a product using "guess and by golly." In the examples
illustrated, the correct answer is always presented as the "last
guess." Some may find this presentation tedious or confusing,
but it is correct -- the author has just decided to show us all of the
WRONG factors first and then presents us with the correct
factor.
WITH PRACTICE, STUDENTS WILL START RECOGNIZING THE
CORRECT FACTORS AND NOT HAVE TO LOOK AT EACH WRONG FACTOR! Please
study the examples, page 287-288, but don't let the author's
presentation intimidate you.
Try factoring trinomials in the form ax2+bx+c
by finding combinations of factors of ax2 and c until the
middle term of bx is obtained when checking. Remember, always pull
out any GCF first and be sure to include GCF with your final
answer. If a trinomial has no GCF, then the terms of its binomial
factor will not contain a common factor (other than 1) either.
Please review examples, pages 288-190. Do not let
the presentation bother you -- the author will show us all the ways that
DON'T work before she shows us the one that DOES work.
With practice, you will not look at every situation that
does not work -- you will start seeing the patterns that do work.
There is another way to factor trinomials -- that is presented the next
section, but it is based on the factoring by grouping that we discussed
at the beginning of this unit.
Factoring Trinomials in the Form ax2+bx+c by
Grouping
Another method that can be used is to apply the
factoring by grouping or grouping method we looked at
earlier. Some treat this as a great "secret" that
makes factoring simple -- I am sure that it will be very beneficial to
some students. IT DOES NOT MATTER WHAT METHOD WE USE TO FACTOR
TRINOMIALS AS LONG AS WE GET AN ANSWER THAT CORRECTLY CHECKS.
In my experience teaching this class, students that
practice the problems and procedures in section 4.3 will start to see
patterns in numbers and will be able to factor trinomials in the form ax2+bx+c
by realizing those factors that will work.
Many of those students will find section 4.4 a bother --
they do not need another method to factor trinomials. Some
students, however, will find that the grouping method makes more sense,
especially if they they did not find the presentation in chapter 4.3
helpful.
I am going to ask students to look at both and apply the
method that works best for them. I will not require anyone to use
one method or the other. Please look at each of these methods and
decide what one will be most comfortable for you.
Factoring Trinomials of the form ax2+bx+c by
grouping:
-
Factor out a greatest common factor, if there is one
other than 1 (or -1)
-
Find 2 numbers whose product is a*c and whose sum is
b.
-
Write the middle term, bx using the factors from
step 2.
-
Factor by grouping (method illustrated in section
4.1).
Please review the examples, pages 295-296.
Remember, you may choose to factor trinomials in the form ax2+bx+c
by the method that you are most comfortable with.
Factoring Perfect Square Trinomials and the Difference
of Two Squares
Remember the "special products" in chapter 3
-- our author stressed them in that part of the book to prepare us to
factor trinomials. Recall the special product formulas for
squaring binomials: (a+b)2 = a2+2ab+b2
and (a-b)2=a2-2ab+b2.
A trinomial that is a square of a binomial is called a perfect
square trinomial, because x2+6x+9 factors into (x+3)2,
it is a perfect square trinomial.
Likewise, 4x2-12x+4 is a perfect square
trinomial, because it factors into (2x-3)2 .
Remember you can factor trinomials by EITHER of the
methods covered in 4.2 or 4.4. It is important to recognize these
special cases, which are really just extensions of the special products
we looked at in Chapter 3. Seeing these patterns will save you
time and make the Chapter 5, rational expressions, much easier.
Review the examples, pages 301-303.
Factoring the Difference of 2 Squares
Recall from Chapter 3 that we looked at anothwer special
product, the product of the sum and difference of 2 terms a and b:
(a+b)(a-b)=a2-b2.
Just like we applied the special products rules to
factor trinomials, we can reverse the equation above to create another
factoring pattern which will be used to factor the difference of 2
squares
Factoring the difference of squares: a2-b2=(a+b)(a-b).
Please review the examples, pages 303-304
Don't forget that we must always check to see if a
trinomial contains a greatest common factor (GCF) BEFORE trying any
other factorization method.
ALWAYS BE SURE TO FACTOR COMPLETELY! Check to see
if any factors can be factored further. Our textbook author is
going to use many examples where a trinomial factors with one term being
the difference of 2 squares (exampler 15, page 304).
Watch for that situation! She is trying to help us
prepare for Chapter 5, Rational Expressions. If you practice the
problems and develop some comfort with these examples, you will do fine
in the chapters that follow.
4.6 Solving Quadratic Equations by Factoring
In many ways, this section is a review of the previous
section (4.5). Please apply yourself fully to this unit. It
will review and reinforce the main concepts in the chapter and prepare
you for Chapter 5.
If we take a trinomial in the form of ax2+bx+c
and set it equal to zero, it is a quadratic equation. Remember, x2+bx+c
is just a special case where a=1.
Quadratic Equation: an equation that can be
written in the form ax2+bx+c=0 where a, b, and c are real
numbers and a does not equal 0.
Note that our definition DOES allow b and c to equal
zero, hence we can have quadratic equations with only our lead
term. Please review the examples on page 313.
If a quadratic equation is factorable (remember, not all
trinomials are factorable), then we can solve it by applying the zero
factor property.
Zero Factor Property: If a and b are real
numbers and if ab=0, then a=0 or b=0.
This is not new -- we have looked at this before.
This understanding tells us that if the product of 2 numbers is 0, then
at least one of the numbers must be 0. Please look at these
examples:
