Algebra 2: Notes 4.3 & 4.4:
Factoring:
GCF: The greatest common factor is the largest factor that
2
2
3
divides all terms evenly.
15x y  5xy  25x
• Step 1: determine the
GCF and write it down.
2
2
3


15
x
y
5
xy
25
x
Then divide each term
5x 



5
x
5
x
5
x
by the GCF.


Answer: 5x 3xy  y 2  5 x 2
• Step 2: Simplify.
GCF  5x

•

Step 3: Check the result 5x  3xy   5x  y   5x  5x 
2
by using Distributive
Property.
15x2 y  5xy 2  25x3 √
2
(4.3 & 4.4): Factor by Grouping
(For polynomials with 4 or more terms)
Step 1: Change any subtract to
add the opposite and Group the
first 2 terms with parentheses
and the last 2 terms.
Step 2: Take out a GCF for the 1st
group, and for the 2nd group.
Simplify.
Step 3: Factor out the common
parentheses. Simplify.
3x  2 x  9 x  6
3x   2 x    9 x  6
3
3
2
2
3
2


   9 x 6 
3
x

2
x
2
x  2   2    3
 
3
 x   3
 x
Pull out a -3 instead
x 2  3x   2    3 3x   2  
 x 2  3x   2   3  3x   2   
 3x   2    3x   2   3x   2  
 
 
 
2
 3
Step 4: Check with FOIL
x 2  3x   x 2  2    3 3x    3 2 
Answer:
 3x  2   x
3x3  2 x 2  9 x  6 √
(4.3 & 4.4): Factoring Trinomials
• 3 ways to factor trinomials:
– Area method
– Guess and check method
– Table method
(4.3 & 4.4): Factoring Trinomials: Area Method
• Step 1: Write the x2 term and the
constant term in the diagonal
rectangles.
• Step 2: Multiply down this
diagonal.
• Step 3: Ask yourself, what two
numbers can you multiply to get
this and add to get the x term?
• Step 4: Write these numbers in
the empty rectangles.
• Step 5: Factor the GCF out of
each column and each row.
• Step 6: Check your factoring.
4x2+8x+3
=
(2x+3)(2x+1)
3
2x
2x
1
*=12x2
4x2
6x
2x
3
+=8x
1x*12x=12x2
1x*12x=13x
2x*6x=12x2
2x*6x=8x
3x*4x=12x2
3x*4x=7x
2x(2x)+2x(1)+3(2x)+3(1)=4x2+8x+3√
(4.3 & 4.4): Factoring Trinomials: Guess & Check
• Step 1: Draw parentheses
• Step 2: Fill in the missing
factors to get the first term.
• Step 3: Fill in the missing
factors to get the last term.
• Step 4: Check by FOIL.
x2 + 10x + 25
(
)(
)
(x
)(x
)
(x + 5)(x + 5)
x(x)+x(5)+5(x)+5(5)
x2 + 10x + 25 √
(4.3 & 4.4): Factoring Trinomials: Table Method
• The Table Method incorporates both the Area
Method and the Guess and Check Method.
4x2+15x+9
=(4x+3)(x+3)
Step 1: Draw parentheses
Step 2: Fill in the missing
factors to get the first term.
Step 3: Fill in the missing
factors to get the last term.
(
)(
( 2x
)(2x
) or ( 4x
)(1x
)
( 2x
)(2x
) or ( 4x
)(1x
)
+1
)
+9
+1
+9
+3
+3
+3
+3
Step 4: Check by FOIL, if it
doesn’t work retry steps 1 ( 4x +3)(1x+3)=4x(x)+4x(3)+3(x)+3(3)=4x2+15x+9√
and 2.
(4.3 & 4.4): Factoring Formulas:
• Difference of Squares
• a2–b2=(a+b)(a–b)
(only works for something squared
minus something squared)
• Sum of Cubes
(only works for something cubed plus
• a3+b3=(a+b)(a2–ab+b2)
something cubed)
• Difference of Cubes (only
works for something cubed minus
something cubed)
• a3–b3=(a–b)(a2+ab+b2)
(4.3 & 4.4): Summary of Factoring:
• Step 1: If all terms have a greatest common factor
other than one, then factor is out.
• Step 2: If the polynomial has:
– Four or more terms, then try the factoring by grouping
method.
– Three terms, then try the guess and check method.
– Two terms, then try factoring using the difference of
squares method, sum of cubes method, or difference of
cubes method.
(4.3 & 4.4): Solve by Factoring:
• Step 1: Factor the polynomial.
• Step 2: Set each factor equal to zero and solve.
Example:
2 x  x  21  0
 2 x  7  x  3  0
2
2x  7  0
7 7
2x
7

2
2
7
x
2
x3 0
3 3
x  3
7
x  or x  3
2
(4.3) √Points: Solve by Factoring:
(4.4) √Points: Solve by Factoring:
(4.3 & 4.4): Factoring Foldable Example:
(4.3) √Points: Solve by Factoring (answers):
(4.4) √Points: Solve by Factoring (answers):