Transcript Factoring Methods

Factoring Methods
Powerpoint created by Linnea Tucker
Factoring with
Algebra Tiles
Factor 2x2 + 5x + 3
•
Using algebra tiles, build a rectangle containing the tiles specified in this
problem (2 x2-tiles, 5 x-tiles and 3 1-tiles). Remember that the lines between
the tiles within your pattern must be completely vertical or horizontal across the
entire pattern. Here is one possible arrangement:
•
After the pattern is established, it can be seen that the top edge of the pattern
(the length) is composed of tiles with dimensions 2x + 3. The side edge of the
pattern (the width) is composed of tiles with dimensions x + 1.
Consequently, 2x2 + 5x + 3 = (2x + 3)(x + 1).
Factoring Trinomials
Step 1 (ALWAYS!)
Can you factor out a Greatest
Common Factor (GCF)?
Example:
GCF = x
2x3 -7x2 + 6x
So, factor it out:
x ( 2x2 – 7x + 6))
Multiply (x + 2)(x + 3)
• Use FOIL
x2 + 3x + 2x + 6
x2 + 5x + 6
Notice the
Multiples of 6
Must add to 5
Factor Cracker
Factor Cracker
Factor
a
2
c
a 2 b
7 x ++ 66
22xx – 7x
2 up the
Stepa1:=set
b =table
-7
c=6
c6
–b7
Factor Cracker
Factor 2 x2 – 7 x + 6
2
6
Factors of
“a” go here
Factors of
“c” go here
–7
Factor Cracker
Factor 2 x2 – 7 x + 6
2
6
2
6
1
1
Factors of 2:
2 and 1
–7
Factors of 6:
1 & 6; -1 & -6;
2 & 3; -2 & -3
(Use trial and error
for placement)
Factor Cracker
2
2
1
Factor 2 x2 – 7 x + 6
Multiply on
6
The
sum
of
the
the diagonal;
right column
products
go
6 6
must
equal
here
+
the “b” term.
1 2
–7
It’s important to select the proper
sign for the factors you use.
2+ bx+ c
x
a
 If the “b” term is negative and the “c” term is
positive, the “b” term factors must both be
negative.
 If the “b” and “c” terms are both negative, then
the “b” term factors must have opposite signs
such that the right hand column produces a sum
that agrees with the “c” term sign.
Factor Cracker
2
2
1
Factor 2 x2 – 7 x + 6
The
sum
ofisn’t
the
If the
sum
If that doesn’t
6
right
column
correct,
swap
work, try the
must
equal
factors
and of
try
-1
-1
other factors
-6 -6
“b”and
term.
+ the
the “a”
“c”
again.
terms.
-1
-6 -2
-12
=
–7
Factor Cracker
2
2
1
Factor 2 x2 – 7 x + 6
The
If
This
that
other
doesn’t
6
factorsswap
work,
combination
of 6
are 2 & 3;
factors.
works!!
-2
-3 -3
-2
+ -2 & -3
-3
-2 -4
-6
=
–7
Factor Cracker
Factor 2 x2 – 7 x + 6
The
Usefactors
FOIL toof
2
6
confirm
the that
the
polynomial
factors are
( 2 x -3 ) -3
correct!
are:
( 1 x -2 ) -4
( 2x-3 ) ( x-2 )
–7
You Try These Ones:
1) Factor
2
x + x - 30
2) Factor 3x2 + 10x +8
3) Factor
2
x – 3x -18
(x+6)(x-5)
(3x+4)(x+2)
(x+3)(x-6)
4) Factor 10x2 - 27x +18
(5x-6)(2x-3)
Box Method
Factor: 2 x2 – 7x + 6
The
quadratic
2
term goes
here
Sums of the
linear term
go here
And here
The constant
term goes
here
2x
6
BoxaMethod
b
c
Factor: 2 x2 – 7x + 6
2x2
Sums of the
linear term
go here
-4x
6
And here
-3x
How do you
determine the
two linear terms?
Step 1: multiply
a·c
2·6 = 12
Step 2: find
factors of a·c that
sum to b.
-3 and -4
Box Method
Factor: 2 x2 – 7x + 6
To the left, list
2x
-3
3
x
2x2
-3x
-2
2
-4x
6
the GCF for each
row.
On the top, list
the GCF for each
column.
Each term takes
on the sign of the
box closest to it.
Box Method
Factor: 2 x2 – 7x + 6
2x
-3
3
x
2x2
-3x
-2
2
-4x
6
The
Factors
are:
(x-2)(2x-3)
Use FOIL to check your answer!!
You Try These Ones:
1) Factor
2
x + x - 30
2) Factor 3x2 + 10x +8
3) Factor
2
x – 3x -18
(x+6)(x-5)
(3x+4)(x+2)
(x+3)(x-6)
4) Factor 10x2 - 27x +18
(5x-6)(2x-3)