Transcript Document

In this course we will study a number of
factoring techniques used to identify the
factors of certain polynomials.
They are:
1.
2.
3.
4.
5.
Greatest Common Monomial Factor
Grouping
Difference of Two Squares
Trinomial type 1 (x2 + bx + c)
Trinomial type 2 (ax2 + bx + c)
An important facet in learning how to factor is
identifying when to apply each technique. The
biggest clue to this is in the number of terms
that the original polynomial contains. We will
focus on this issue later when we have learned
several techniques.
Greatest Common Monomial Factor
Let’s break this down to identify exactly what
this is. First of all a factor is basically a
divisor because all of those numbers can divide
evenly into it.
Factors of 12  1, 2, 3, 4, 6, 12
Factors of 18  1, 2, 3, 6 ,9, 18
Common Factors for 12 and 18  1, 2, 3, 6
Greatest Common Factor for 12 and 18  6
A monomial factor is a one term factor. This
expression is sometimes used when finding factors of
polynomials.
In knowing your times table, you know the factors of many, many
numbers virtually instantaneously. If you don’t know your times
table very well, then it can be very time-consuming when you want
to determine the factors of numbers.
It is very useful to know the prime factors of a given number.
This refers to the prime numbers that are multiplied to produce
the given number.
Examples:
15 = 3 × 5
12 = 2 × 2 × 3
16 = 2 × 2 × 2 × 2
75 = 3 × 5 × 5
39 = 3 × 13
42 = 2 × 3 × 7
84 = 2 × 2 × 3 × 7
112 = 2 × 2 × 2 × 2 × 7
Notice that some of the given numbers are expressed with 2
factors, others with 3 factors and yet others with more than 3.
All of the factors are prime numbers. Prime numbers are the most
basic factors you can get.
An easy way to determine the prime factors of a number is by
constructing a factor tree where the given number is the trunk of
the tree and the prime factors are the ends of the branches.
72
72
8
2
9
4
3
6
3
2
12
3
2
6
2 × 2× 2× 3 × 3
2 × 3 × 2 × 2 × 3
Therefore: 72 = 2 × 2 × 2 × 3 × 3
Notice that no matter which two factors that you begin with, you
end up with the same prime factors.
225
15
15
3 × 5× 3 × 5
Therefore: 225 = 3 × 5 × 3 × 5 OR 225 = 3 × 3 × 5 × 5
Just as numbers have factors so do polynomial expressions.
The factors of monomial expressions are pretty obvious, however
the factors of polynomial expressions having 2 or more terms are
not so obvious. For this reason it is necessary to learn techniques
that illuminate the factors of these kinds of polynomial
expressions.
Monomial expressions
2x
=2x
108p2q = 2  2  3  3  3  p  p  q
24a2bc3 = 2  2  2  3  a  a  b  c  c  c
Polynomial expressions with 2 or more terms
Given the polynomial expression:
24x4y3z7 + 16x2y5z4 - 44xz6
Each of these 3 terms have many factors.
They have factors that are common to all 3 terms –
many, in fact. Ex: 1, 2, 4, x, z, xz, 4z2, xz3, 4xz4
However, they have only 1 greatest common factor
which is a monomial term. (4xz4)
We can divide each of the terms of the polynomial by
this GCF.
24x 4 y 3z 7
3 3 3

6
x
y z
4
4xz
16x2 y 5z 4
5

4
xy
4xz4
44xz6
2

11
z
4xz4
24x 4 y 3z 7
3 3 3

6
x
y z
4
4xz
16x2 y 5z 4
5

4
xy
4xz4
44xz6
2

11
z
4xz4
By dividing each of the terms of the polynomial by
this GCF, rewrite the equivalent value of the
polynomial as 2 separate factors. One factor being
the GCF monomial and the other being the leftover
terms after division.
Leftover
terms
( 6x3y3z3 + 4xy5 - 11z2 )
GCF
4xz4
If we multiply 4xz4 by 6x3y3z3 + 4xy5 - 11z2 then we
will arrive back at the polynomial that we started
with.  24x4y3z7 + 16x2y5z4 - 44xz6
This is a way to verify that we have correctly
factored the polynomial.
Factor 3x2 + 6xy
When determining the GCF of a polynomial, we can
take it in steps.
1. Determine the GCF of the numerical coefficients.
GCF of 3 and 6 is 3
2. Determine the GCF of each variable. We can do
this by choosing the lowest power of a given variable
from each term.
GCF of x2 and x is x because x is the
variable that has lowest exponent of those
2 (exponent is 1).
With no y-variable in the first term and y in
the second term there can’t be a common
factor containing variable y.
3x2
Factor:
+ 6xy
GCF = 3x
Factors: 3x(x + 2y)
3x2
x
3x
6xy
 2y
3x
Factor: 10x4y3 + 4x3y2 - 2x2y2
GCF of 10, 4 and 2 is 2
GCF of x4, x3 and x2 is x2 because x2 is the
variable that has lowest exponent of those
3 (exponent is 2).
GCF of y3, y2 and y2 is y2 because y2 is the
variable that has lowest exponent of those
3 (exponent is 2).
GCF = 2x2y2
Factor: 10x4y3 + 4x3y - 2x2y2
10x 4 y3
2

5
x
y
2 2
2x y
4x3y2
 2x
2 2
2x y
GCF = 2x2y2
2x2 y2
1
2 2
2x y
Factors: 2x2y2(5x2y2 + 2x - 1)
Factor: -3m3n – 7m3r + 8m3rt
GCF of 3, 7 and 8 is 1
GCF of m3, m3 and m3 is m3 because each
term has the variable m3.
Only one term has the variable n so it is not
common to all terms
Not all terms have the variable r so it is
not common to all terms
Only one term has the variable t so it is not
common to all terms
Factor: -3m3n – 7m3r + 8m3rt
GCF = 1m3
Because the leading term (-3m3n) is negative, we
will make the common factor negative. GCF = -1m3
 3m3n
 7 m3r
 3n
 7r
3
3
 1m
 1m
Factors: -m3(3n + 7r – 8rt)
8m3rt
 8rt
3
 1m
Factor: -2ab3 – 4b3c – 12b3d
GCF of 2, 4 and 12 is 2
Only one term has the variable a so it is not
common to all terms
GCF of b3, b3 and b3 is b3 because each
term has the variable b3.
Not all terms have the variable c so it is
not common to all terms
Only one term has the variable d so it is
not common to all terms
Factor: -2ab3 – 4b3c – 12b3d
GCF = -2b3
Because the leading term (-2ab3 ) is negative, we
will make the common factor negative. GCF = -2b3
 2ab3
 4b3c
a
 2c
3
3
 2b
 2b
Factors: -2b3 (a + 2c + 6d)
 12b3d
 6d
3
 2b