Transcript Factoring Expressions

Factoring Expressions
Review
Definitions and Key Ideas
Factoring is the process of writing an
expression as a multiplication problem.
We can write numbers in terms of factors:
What are the factors of 12?
Factoring
“undoes”
multiplication
Answer  1, 2, 3, 4, 6, 12
To factor 12, we would write it as a
product of its factors:
2 23
These are special factors
of 12 known as prime
factors.
What is
special
about prime
factors?
Algebraic Expressions
There are several different methods used to
factor algebraic expressions, or expressions
that contain variables.
All of our factoring methods are built around
the idea of area of a rectangle.
AreaRectangle =
(base)(height)
Because factoring is the process of
“undoing” multiplication, let’s review
our multiplication techniques.
Multiplication Review
Example 1: 4(2 x  3)
There are 2 methods used to multiply
these expressions.
Distributive Property
Area and Rectangles
2x + 3
Multiply the inside terms by
the outside value.
4( 2 x  3)
A = bh
4
8x
12
4  2x  4  3
8x  12
The total area of the large
rectangle is 8x + 12.
Factoring
How can we work backwards
from this process?
Think about
the algebra
tiles– can we
build a
rectangle?
Let’s look at the
dimensions of
the rectangle we 4
made.
Simplify the
base!
We need to start with
our solution: 8x  12
x
x
1 1 1
2x
+
3
Area = (base)(height)
** Don’t forget that
using generic
rectangles can save
time!
8x + 12 = (2x + 3)(4)
Factoring
Example 2: Using Generic Rectangles
4 x  8x
2
When asked to factor, you
are given the area and are
asked to find the
dimensions.
1) Find a height that
works for both small
rectangles.
2) Use that height and
the areas to find the
base.
The factored form:
4x(x+2)
x
4x
4x 2
+
2
8x
When finding the dimensions you
want to find the largest height
possible. This height is the
GREATEST COMMON FACTOR!
Practice
Use the concept of area to find
the factored form of each of the
following expressions.
Solutions:
1)3x  6
Once you have
1
)
3
(
x

2
)
completed
all
2)12 x  18
the problems,
2)6(2 x  3)
2
click the mouse
3)5 x  10 x
again for the
3)5 x( x  2)
3
4)3x  9 x
solutions!
2
4)3x( x  3)
Multiplication Review
Example 2
( x  5)( x  3)
•Again we are going to
use the concept of area to
multiply our 2 factors.
x
Don’t
•Use each of the
factors for either the forget to +
combine
base or the height.
5
like
•Multiply base and
height to find the area. terms!
( x  5)( x  3)  x 2  8x  15
x
+
3
2
3x
5x
15
x
** Because it is a multiplication
problem, it does not matter which
factor is the base and which is the
height.
Factoring
Remember that factoring is
working backwards to write a
multiplication problem.
When given an expression with 3 terms, there are 2
methods of factoring that can be used.
2
x
 9 x  20
Method 1: Generic
Rectangle
x
x

4

5
x2
5x
4x
20
4x  ___
5x  9 x
___
We use 4 & 5
because they also
multiply to equal
20!
Once all of the areas are
labeled, we have to find
the dimensions.
Label each of the
individual rectangles,
then find the dimensions
of the larger rectangle.
x 2  9 x  20  ( x  5)( x  4)
Factoring
A
M
x 2  9 x  20
Method 2: Diamond Problems
Note in the previous
example that in order to
fill in the areas for the
smaller rectangles, we
needed to find 2 numbers
that added up to 9 and
multiplied to 20.
Anytime you have an
expression with 3 terms and
there is not a coefficient with x 2
we can use a diamond problem
to find our factors.
20
This is the same process 4
5
we used to complete
9
diamond problems.
x 2  9 x  20  ( x  5)( x  4)
Practice
Use either generic rectangles or diamonds to
factor each of the following.
1) x 2  6 x  8
2) x 2  8 x  12
3) x 2  2 x  15
4) x 2  3 x  18
Once you
have factored
all of these,
click the
mouse again
to check your
answers!
1)( x  2)( x  4)
2( x  6)( x  2)
3)( x  5)( x  3)
4)( x  6)( x  3)
Remember, the order of
the factors does not
matter!
Special Cases
There are 2 special cases for factoring: Difference
of Squares and Perfect Square Trinomial.
Difference of Squares
Example:
Difference of squares is used
when you have 2 terms
separated by subtraction.
x  16
2
x 2  0 x  16
2) Use a diamond problem to factor.
-16
-4
4
0
1) Rewrite the expression with 3 terms.
3) Write final factors.
( x  4)( x  4)
Special Cases
Perfect Square Square Trinomials
x 2  12 x  36
A perfect square
trinomial looks just
like a diamond
problem. The
difference is in how
we write the answer.
36
6
6
12
( x  6)( x  6)
( x  6) 2
1) Factor using a
diamond.
2) What do you
notice about
your solution?
Can we write it
a simpler way?
Practice
In each of the problems below, first decide whether
you have a difference of squares or a perfect square
trinomial, then factor.
1) x  9
2
2) x 2  9 x  20
3) x  x  20
2
4) x 2  25
Solve each of
the problems.
After you
finish, click
the mouse to
check your
answers.
1)( x  3)( x  3)
2)( x  4)( x  5)
3)( x  5)( x  4)
4)( x  5)( x  5)
Now that we have talked about several methods
of factoring, let’s put them together!!
Factoring Completely
Factoring completely is a combination of
generic rectangles and diamond problems
to present our answer in simplest form!
2 x 4  8x 2
x2  4
1) Set up a generic rectangle to factor the
GCF– find the largest height!
2
2x
2) Find the base of the rectangle.
3) Try to write the base
with 3 terms to factor.
4) Now, we can use a
diamond to factor the base.
5) Write your final expression.
2x
4
8x 2
2 x 2 ( x 2  4)
2 x 2 ( x 2  0 x  4)
-4
2
-2
0
2 x ( x  2)( x  2)
2
Note that the terms in
the example do not have
anything in common, but
there is still a coefficient
(3) in front of x2.
An Exception– ax2
Not all expressions have a GCF.
3x  7 x  2
2
We need to use a modified
diamond.
1. Multiply the 1st and 3rd terms.
2. Set up a diamond problem
6
with the product in the top.
6
1
3. Rewrite the expression with
7
4 terms.
3 2  6
3x  6 x  1x  2
2
4. Use these terms to fill in the
area of a generic rectangle.
x
2
3x
3x 2
6x
5. Find the dimensions and
write the factors.
1
1x
2
(3x  1)( x  2)
Practice
Factor each of the polynomials completely. Don’t
forget to start with the GCF (largest height) first!
1)3 x  24 x  45
2
2) 4 x 2  4
3)5 x 2  13 x  6
4) 2 x 2  12 x  18
After you
factor each
problem,
click the
mouse again
to check your
answers.
1)3( x  3)( x  5)
2)4( x  1)( x  1)
3)(5 x  2)( x  3)
4)2( x  3) 2
Review
2 Terms:
Difference
of Squares
Remember: In order to factor, start with a
GCF (largest height) first. Then, look at
the number of terms in the base to decide
where to go next!
3 terms:
Modified
Diamond
Diamond
Problem
3 x 2  75
3( x  25)
2
3( x 2  0 x  25)
3( x  5)( x  5)
5x 2  7 x  6
5  6  30
5 x 2  10 x  3x  6
(5 x  3)( x  2)
2 x 2  6 x  36
2( x 2  3x  18)
2( x  6)( x  3)