Of all the topics covered in this chapter factoring polynomials is
probably the most important topic.
There are many sections in later chapters where the first step will be to
factor a polynomial. So,
if you can’t factor the polynomial then you won’t be able to even start the problem let alone finish it.
Sumber
Let’s start out by talking a little bit about just what factoring
is. Factoring is the process by which we go about determining what we multiplied to get the given quantity.
We do this all the time with numbers.
For instance, here are a variety of ways to factor 12.
There are many more possible ways to factor 12, but these are representative
of many of them. A common method
of factoring numbers is to completely factor the number into positive
prime factors. A prime
number is a number whose only positive factors are 1 and itself. For
example 2, 3, 5, and 7
are all examples of prime numbers. Examples of numbers that aren’t prime are 4,
6, and 12 to pick
a few.
If we completely factor a number into positive prime factors there
will only be one way of doing it. That is the reason for factoring
things in this way. For our example above with 12 the
complete factorization is,
12 = (2)(2)(3)
Factoring polynomials is done in pretty much the same manner. We
determine all the terms that were
multiplied together to get the given polynomial. We then try to factor each of
the terms we found in the
first step. This continues until we simply can’t factor anymore. When we can’t
do any more
factoring we will say that the polynomial is completely factored.
Here are a couple of examples.
x2 −16 = (x + 4)(x − 4)
This is completely factored since neither of the two factors on the
right can be further factored. Likewise,
x4 −16 = (x2 + 4)(x2 − 4)
is not completely factored because the second factor can be further
factored. Note that the first factor is completely factored however. Here is the complete factorization
of this polynomial.
x4 −16 = (x2 + 4)(x + 2)(x − 2)
The purpose of this section is to familiarize ourselves with many
of the techniques for factoring polynomials.
Greatest Common Factor
The first method for factoring polynomials will be factoring out
the greatest common factor. When factoring in general this will also be the first thing that we
should try as it will often simplify
the problem.
To use this method all that we do is look at all the terms and
determine if there is a factor that is in common to all the terms. If there is, we will factor it out of the
polynomial. Also note that in this case we are really only using
the distributive law in reverse. Remember that the distributive law states that
a (b + c) = ab + ac
In factoring out the greatest common denominator we do this in
reverse. We notice that each term has an a in it and so we “factor” it out using the
distributive law in reverse as follows,
ab + ac = a(b + c)
Factoring By Grouping
This is a method that isn’t used all that often, but when it can be
used it can be somewhat useful.
Factoring by grouping can be nice, but it doesn’t work all that
often. Notice that as we saw in the last two parts of this example if there is a “-” in front of the
third term we will often also factor that out of the third and fourth terms when we group them.
Factoring Quadratic Polynomials
First, let’s note that quadratic is another term for second degree polynomial.
So we know that the largest
exponent in a quadratic polynomial will be a 2. In these problems we will be
attempting to factor
quadratic polynomials into two first degree (hence forth linear) polynomials.
Until you become good at
these, we usually end up doing these by trial and error although there are a couple of processes that can make
them somewhat easier.
Special Forms
There are some nice special forms of some polynomials that can make
factoring easier for us on occasion.
Here are the special forms.
a2 + 2ab + b2 = (a
+ b)2
a2 – 2ab + b2 = (a – b)2
a2 –b2 = (a + b)(
a – b)
a3 + b3 = (a
+ b)(a2 – ab + b2)
a3 – b3 = (a –
b)(a2 + ab + b2)
Do not make the following factoring mistake!
a2 + b2 ≠ (a + b)2
This just simply isn’t true, so be careful not to make this very
common mistake. The sum of two perfect
squares doesn’t factor!
Factoring Polynomials with Degree Greater than 2
There is no one method for doing these in general. However, there
are some that we can do so let’s take a look at a couple of examples.
We did not do a lot of problems here and we didn’t cover all the
possibilities. However, we did cover some of the most common techniques that we are liable to run
into in the other chapters of this
work.
Sumber
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