Factoring
Common Mistakes
Factoring-Greatest Common Factor
How to Find the GCF
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To find the GCF look at all
terms and break up each
term into its multiples.
Find all factors of each term
that are in common.
Multiply all common factors
to form the GCF.
Factor the GCF from the
polynomial.
Complete Manual: ..\Factoring Review.docx
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Common Mistakes

Not factoring out all common factors
or factoring GCF from all terms.
Factor the GCF : 3 x 3 y − 6 x 2 y 2 − 9 xy
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Incorrect:
3 x( x 2 y − 2 xy 2 − 3 y )
or 3 xy (3 x 2 − 2 xy − 9 xy )
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Correct:
3 xy (3 x 2 − 2 xy − 3)
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Cancelling out a common factor.
Factor the GCF : 3 x 3 y − 6 x 2 y 2 − 3 xy

Incorrect: 3 xy ( x − 2 xy )

Correct: 3 xy ( x 2 − 2 xy − 1)
2
Factoring-By Grouping
How to Factor by
Grouping
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Not all polynomials have a
greatest common factor.
But there will be terms that have a
variable in common while other
terms have a different variable in
common.
Group the different set of terms
and factor the GCF for each
group.
Factor out the common groupings.
Common Mistakes

Group incorrectly or not factoring
out the common groupings.
Factor by grouping : x 2 y 3 − 2 y 3 − 2 x 2 + 4

Incorrect: x 2 y 3 − 2 y 3 − 2 x 2 + 4
= ( x 2 y 3 + 4) + (−2 y 3 − 2 x 2 )
or ( x 2 y 3 − 2 y 3 ) + (−2 x 2 + 4)
= y 3 ( x 2 − 2) + 2(− x 2 + 2)

Correct:
x2 y3 − 2 y3 − 2x2 + 4
= ( x 2 y 3 − 2 y 3 ) + (−2 x 2 + 4)
= y 3 ( x 2 − 2) − 2( x 2 − 2)
= ( x 2 − 2)( y 3 − 2)
Note: The last line of the correct form is
not factored completely.
Complete Manual: ..\Factoring Review.docx
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Factoring-Difference of Squares
How to Factor Using
Difference of Squares

Use the special product:
a 2 − b 2 = (a + b)(a − b)
Common Mistakes

Determining a or b incorrectly.
Factor completely : 9 x 2 − 4
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Incorrect:
(9 x + 4)(9 x − 4) or (3 x + 4)(3 x − 4)
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Correct:
(3 x + 2)(3 x − 2)
Not factoring completely.
Factor completely : x 4 − 1
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Incorrect:
( x 2 + 1)( x 2 − 1)
Correct:
( x 2 + 1)( x 2 − 1) = ( x 2 + 1)( x + 1)( x − 1)
Complete Manual: ..\Factoring Review.docx
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Factoring-Sum and Difference of Two Cubes
How to Factor Using
Sum/Difference of Cubes

Common Mistakes
Use the special products:
a + b = (a + b)(a − ab + b )
3
3
2
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Determining a or b incorrectly.
Factor completely : 8 x 3 + 1
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Incorrect:
8 x 3 + 1 = (8 x + 1)(8 x 2 − 8 x + 1)
2
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a 3 − b 3 = (a − b)(a 2 + ab + b 2 )
Correct: 8 x 3 + 1 = (2 x ) 3 + (1) 3
= (2 x + 1)(4 x 2 − 2 x + 1)
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Using formula incorrectly.
Factor completely : x 3 − 27
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Incorrect:
x 3 − 27 = ( x − 3)( x 2 − 6 x + 9)
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Correct:
x 3 − 27 = ( x − 3)( x 2 + 3 x + 9)
Complete Manual: ..\Factoring Review.docx
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2
Factoring-Trinomials of the form x + bx + c
How to Factor Trinomials
where a is 1
 Integers are all the whole
numbers and there
negatives:
…-3,-2,-1,0,1,2,3,…
 To factor trinomials of the
form x 2 + bx + c find two
integers whose sum equals
the middle term and whose
product equals the last term.
Common Mistakes
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Sum is not the middle term and/or
whose product is not the last term
Factor completely : x 2 − 5 x + 6
2
Incorrect: x − 5 x + 6 ≠ ( x − 6)( x + 1)
Check : ( x − 6)( x + 1) = x 2 + x − 6 x − 6
= x 2 − 5x − 6 ≠ x 2 − 5x + 6
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Correct: x 2 − 5 x + 6 = ( x − 3)( x − 2)
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If c = 0, using b for the last term.
Factor completely : x 2 − 10 x
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2
Incorrect: x − 10 x ≠ ( x − 5)( x + 2)
Check : ( x − 5)( x + 2) = x 2 + 2 x − 5 x − 10
= x 2 − 3 x − 10 ≠ x 2 − 10 x

Complete Manual: ..\Factoring Review.docx
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Correct:
x 2 − 10 x = x( x − 10)
2
Factoring-Trinomials of the form ax + bx + c
How to Factor Trinomials
where a is not 1
 To factor trinomials of the
2
form ax + bx + c find two
integers whose sum equals
the middle term and whose
product equals the product
ac.
 Use the two integers to
break up the trinomial and
factor by grouping.
Complete Manual: ..\Factoring Review.docx
To view; right click and open the hyperlink
Common Mistakes

Sum is not the second term and/or
whose product is not the last term.
Factor completely : 2 x 2 − 7 x − 4
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Incorrect:
2 x 2 − 7 x − 4 ≠ (2 x − 2)( x + 2)
Check : (2 x − 2)( x + 2) = 2 x 2 + 4 x − 2 x − 4
= 2x2 + 2x − 4 ≠ 2x2 − 7x − 4
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Correct:
2 x2 − 7 x − 4 = 2 x2 − 8x + x − 4
= 2 x( x − 4) + ( x + 4)
= ( x + 4)(2 x + 1)