Transcript Solving by Factoring Remediation Notes

Ways to factor…
There a few different ways to approaching factoring an
expression. However, the first thing you should always
look for is the Greatest Common Factor (GCF).
How to find the GCF video
Factoring using GCF video
Why GCF first?
You look for the GCF first because it will help you factor
quadratics using the second method by making the
numbers smaller.
The second method of factor involves undoing the
distributive property….I call it unFOILing 
There are several videos posted on this method, but
here’s an example…..
Factor: x2 + 6x + 8
Look at the last number.
If the sign in a positive, the signs in the parenthesis
will be the same.
x2 + 6x + 8
Here the 8 is positive.
Look at the sign on the middle number.
We know the signs will be the same because 8 is
positive. We look a the middle number and it's also
positive. So both signs in the parenthesis will be
positive.
(x + )(x + )
Find factors of the last number that when you
mulitply them you get that last number, but when
you combine them you get the middle number.
So we're looking for factors of 8 that we multiply
them we get an 8, but when we add them we get a
6.....4 and 2.
(x + 4)(x + 2)
Check it with FOIL.
You never get a factoring problem wrong! You can
always check it by multiplying.
(x + 4)(x + 2) = x2 + 4x +2x +8
It works!
Factor: x2 - 7x + 12
Look at the last number.
If the sign in a positive, the signs in the parenthesis will be the same.
x2 - 7x + 12
Here the 12 is positive.
Look at the sign on the middle number.
We know the signs will be the same because 12 is positive. We look a the
middle number and it's negative. So both signs in the parenthesis will be
negative.
(x - )(x - )
Find factors of the last number that when you mulitply them you get
that last number, but when you combine them you get the middle
number.
So we're looking for factors of 12 that we multiply them we get an 12, but
when we add them we get a 7.....4 and 3.
(x - 4)(x - 3)
Check it with FOIL.
You never get a factoring problem wrong! You can always check it by
multiplying.
(x - 4)(x - 3) = x2 - 4x - 3x + 12
= x2 - 7x + 12
Factor: x2 - 3x - 54
Look at the last number.
If the sign in a negative, the signs in the parenthesis will be
opposites.
Look at the sign on the middle number.
Since we know we will have one of each, the middle
number now gives the sign to the greater factor.
Find factors of the last number that when you mulitply
them you get that last number, but when you combine
them you get the middle number.
So we're looking for factors of 54 that we multiply them to
a get a 54, but when we subtract them we get a 3.....9 and
6.
Check it with FOIL.
You never get a factoring problem wrong! You can always
check it by multiplying.
x2 - 3x - 54
Here the 54 is negative.
(x - )(x + )
(x - 9)(x + 6)
(x - 9)(x + 6) = x2 - 9x + 6x - 54
= x2 - 3x - 54
Special Case: The Difference of two Perfect Squares
The difference of two perfect squares is very easy to
factor, but everyone always forgets about them.!They're
in the form (ax)2 - c where a and c are perfect squares.
There's no visible b-value...so b = 0. You factor
them by taking the square root of a and the square root
of c and placing them in parenthesis that have opposite
signs.
Whenever you have a binomial that is subtraction, always check to
see it’s this special case. It usually does NOT have a GCF.
Here's an example….
Example
Factor : 4x2 – 9
Set up parenthesis with opposite signs
( + )( - )
Find the square root of a and place then
answer in the front sections of the
parenthesis
sqrt(4x2) = 2x
( 2x + )( 2x - )
Find the square root of c and place them
at the end of the parenthesis.
( 2x + 3 )( 2x - 3 )
sqrt(9) = 3
Difference of Two Perfect Squares Video
Practice Factoring
1. x2 + 4x – 5
2. x2 - 3x + 2
3. x2 - 6x – 7
4. x2 + 4x + 4
Solutions
1. x2 + 4x – 5 = (x+5)(x-1)
2. x2 - 3x + 2 = (x-1)(x-2)
3. x2 - 6x – 7 = (x-7)(x+1)
4. x2 + 4x + 4 = (x +2)(x+2)
Practice:
Common
Factors
Practice:
Difference
of Two
Squares
Practice:
Factor
a=1
Factoring
with
Algebra
Tiles
What if the leading coefficient isn’t a 1?
Factor: 3x2 + 11x - 4
Set up two pairs of parenthesis
Look over the equation
Look at the a-value
Unfortunately, the a-value is not a one, so we need
to list factors in a chart.
We're looking for the pair of factors that when I
find the difference of the products
will yield the b-value.
(
)(
)
( +
)( - )
Factors of A Factors of C
1, 3
2,2 and 1,4
1*2 - 3*2 = -4 NO
1*3 - 1*4= -1 NO
1*1 - 3*4= -12 YES!
Enter in values
(x - 4)(3x + 1)
Check with FOIL
(x - 4)(3x + 1)= 3x^2 -12x + x - 4
It's possible that you have the right numbers but in
= 3x^2 -11x -4
the wrong spots, so you have to check.
Factoring when a≠ 1
Terms in a quadratic expression may have some common factors
before you break them down into linear factors.
Remember, the greatest common factor, GCF, is the
greatest number that is a factor of all terms in the expression.
When a ≠ 1, we
should always check to see if the quadratic expression has a
greatest common factor.
Factor
2
2x -22x
+36
Step 1:
a ≠ 1, so we should check to see if the quadratic expression
has a greatest common factor.
It has a GCF of 2.
2x2 -22x +36 = 2(x2 -11x +18)
Step 2:
Once we factor out the GCF, the quadratic expression now has a value of
a =1 and we can use the process we just went through in the previous
examples.
x2 -11x +18 = (x -2)(x-9)
Therefore, 2x2 -22x +36 is = 2 (x -2)(x-9).
A≠ 1 and NO GCF
2x2 + 13x – 7
 Step 1: a ≠ 1, so we should check to see if the quadratic
expression has a greatest common factor.
It does not have a GCF!
 This type of trinomial is much more difficult to factor
than the previous. Instead of factoring the c value
alone, one has to also factor the a value.
 Our factors of a become coefficients of our x-terms and
the factors of c will go right where they did in the
previous examples.
2x2 + 13x – 7
Step 1: Find the product ac.
ac= -14
Step 2: Find two factors of ac that add to give b.




􀀹 1 and -14 = -13
􀀹 -1 and 14 = 13 This is our winner!
􀀹 2 and -7 = -5
􀀹 -2 and 7 = 5
Step 3: Split the middle term into two terms, using the
numbers found in step above.
2x2 -1x + 14x – 7
Step 4: Factor out the common binomial using the box
method.
2x2 -1x + 14x – 7
Quadratic
Term
Factor 1
2x2
-1x
Factor 2
Constant
Term
14x
-7
Find the GCF for each column
and row!
Numbers in RED represent the GCF
of each row and column
x
7
2x
-1
2
2x
-1x
14x
-7
The factors are (x + 7)(2x - 1).
Practice Factoring
2
1. 2x 11x + 5
2
2. 3x - 5x - 2
3. 7x2 - 16x + 4
2
4. 3x + 12x + 12
Solutions
1.
2.
3.
4.
2
2x
+11x + 5 = (2x + 1)(x + 5)
2
3x - 5x - 2 = (3x + 1)(x - 2)
7x2 - 16x + 4 = (7x - 2)(x - 2)
3x2 + 12x + 12 = 3(x + 2)(x + 2)
Special Products
Difference of Squares
x2 - y2 = (x - y) (x + y)
Square of Sum
x2 + 2xy + y2 = (x + y)2
Square of Difference
x2 - 2xy +y2= (x - y)2
Factoring Strategies
Is there a
GCF?
2 Terms…
3 Terms…
Look for
special
products.
Look for
squares of a
difference or a
sum.
Prime Factors
Remember:
This won’t work for all quadratic trinomials, because
not all quadratic trinomials can be factored into
products of binomials with integer coefficients.
We call these prime!
(Prime Numbers are 3, 5, 7, 11, 13, etc.)
Expressions such as x2 + 2x - 7, cannot be factored at
all, and is therefore known as a prime
polynomial.
Practicing Factoring when
a ≠1.
Please watch the demonstration below on factoring
when a ≠ 1. There will be interactive examples
provided to help when a ≠ 1.
MORE FACTORING
Upon completion of the video and
demonstration, please complete Mastery Assignment
Part 2.
Gizmo: Factoring
ax2 + bx + c
Practice:
Application
Problems
Practice:
All Other
Cases
More
Instruction