review of factoring methods in this slideshow you will see examples of: 1.factor gcf for any #...
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Review of Factoring Methods
In this slideshow you will see examples of:
1.Factor GCF for any # terms
2.Difference of Squares binomials
3.Sum or Difference of Cubes binomials
4.PST (Perfect Square Trinomial) trinomials
5.Reverse of FOIL trinomials
6.Factor by Grouping usually for 4 or more terms
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Example 1: GCFFIRST STEP for every expression: factor the GCF!
5x3 – 10x2 – 5x
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Example 1: GCFFIRST STEP for every expression: factor the GCF!
5x3 – 10x2 – 5x
5x(x2 – 2x – 1)
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Example 2: GCF
3(x + 1)3 – 6(x + 1)2
Hint: Remember that a “glob” can be part of your GCF.
Do you see a parenthetical expression repeated here?
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Example 2: GCF
3(x + 1)3 – 6(x + 1)2
3(x + 1)2 [ ] this is the GCF
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Example 2: GCF
3(x + 1)3 – 6(x + 1)2
3(x + 1)2 [ (x+1) -2 ]
Left-over factors from: 1st term 2nd term
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Example 2: GCF
3(x + 1)3 – 6(x + 1)2
3(x + 1)2 [ (x+1) -2 ] 3(x + 1)2
Combine like terms: +1 -2
(x – 1)
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Example 3:Difference of Squares
75x4 – 108y2
GCF first! 3(25x4 – 36y2)3(5x2 – 6y) (5x2 + 6y)Recall these binomials are called conjugates.
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IMPORTANT!
Remember that the difference of squares factors into conjugates . . .
However, the SUM of squares is PRIME – cannot be factored!
a2 + b2 PRIME a2 – b2 (a + b)(a – b)
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Example 4:Sum/Difference of Cubes
a3 - b3
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a3 - b3
( ) ( )
Example 4:Sum/Difference of Cubes
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a3 - b3
(a - b) ( )
Cube roots w/ original sign in the middle
Example 4:Sum/Difference of Cubes
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Example 4:Sum/Difference of Cubes
a3 - b3
(a - b) (a2 + b2)
Squares of those cube roots.
Note that squares will always be positive.
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Example 4:Sum/Difference of Cubes
a3 - b3
(a - b) (a2 + ab + b2)The opposite of the product
of the cube roots
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p3 - 125(p - 5)
Cube roots of each Squares of those cube roots &
with same sign opp of product of roots in middle
Example 5:Sum/Difference of Cubes
+ 5p(p2 + 25)
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(4x2 + 9y2)
8x3 + 27y3
(2x +3y)
Cube roots of each Squares of those cube roots &
with same sign opp of product of roots in middle
Example 5:Sum/Difference of Cubes
– 6xy
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Example 6:Difference of Cubes
m6 – 125n3
(m2 – 5n)
Cube roots of each Squares of those cube roots & with same sign opp of product of roots in middle
(m4 + 25n2)+ 5m2n
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Example 7: Special Case * 1ststep: Diff of Squares * 2nd step: Sum/Diff of Cubes
x6 – 64y6
( ) ( ) ( )( ) ( )( )
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Example 7: Special Case * 1ststep: Diff of Squares
* 2nd step: Sum/Diff of Cubes
x6 – 64y6
(x3 – 8y3) (x3 + 8y3)
( )( ) ( )( )
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Example 7: Special Case * 1ststep: Diff of Squares
* 2nd step: Sum/Diff of Cubes
x6 – 64y6
(x3 – 8y3) (x3 + 8y3)
(x–2y)(x2+2xy+4y2) (x+2y)(x2-2xy+4y2)
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Example 8: PST
9x2 – 30x + 25
( ) 2
Recall the PST test:
Are the1st & 3rd terms squares? Is the middle term twice the product of their square roots?
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Example 8: PST
9x2 – 30x + 25
(3x – 5) 2
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Example 9: Reverse FOIL (Trial & Error)
6x2 – 17x + 12
( ) ( ) 3x – 4 2x – 3
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Reverse FOIL(Trial & Error)
Hint: don’t forget to read the “signs”ax2 + bx + c ( + )( + )ax2 – bx + c ( – )( – )ax2 + bx – c ( + )( – )
positive product has larger value
ax2 – bx – c ( + )( – ) negative product has larger value
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Special Case 10: Some quartics can be factored like
quadratics (x4 x2 ● x2)
x4 – 5x2 – 36
( ) ( ) x2 – 9 x2 + 4
But, you aren’t done yet! Do you see why?
(x + 3)(x – 3)(x2 + 4)
Now you’re done!
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Example 11: Factor by Grouping
(4 or more terms)
a(x – 7) + b(x – 7)
(x – 7) (a + b)•Note that this is a BINOMIAL – only two terms here
•Do you see that (x – 7) is a common glob or GCF?
•To factor by grouping, your goal will be to rewrite a statement so it will have such factorable globs!
Glob is the GCF Left-over factors
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Example 11: Factor by Grouping
(4 or more terms)
x3 – 2x2 + ax – 2a•Can you take a GCF out of the first pair and a GCF out of
the second pair?•Will this leave a common GLOB as a GCF?
(If not, rearrange the order of terms & try a different plan.)•We will call this “Grouping 2 X 2”
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Example 11: Factor by Grouping – 2 X 2
x3 – 2x2 + ax – 2a
x2 [x – 2 ] + a [x – 2]
[x – 2 ]Glob is a GCF “Left-Over”Factors
(x2 + a)
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Summary: Factor by Grouping 2 X 2
x3 – 2x2 + ax – 2a Look for two small
[x3 – 2x2] +[ ax – 2a] factorable groups!
x2 [x – 2] + a [x – 2] Check IF same
leftover factor (glob)!
[x – 2 ] (x2 + a) Pull the final GCF out in front of the
leftover factors .
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Example 12: Factor by Grouping – 2 X 2
m2 – n2 + am + an
[m – n] [m + n] +
[m + n]Glob is a GCF “Left-Over”Factors
([m – n] + a)
a [m + n]
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Summary: Factor by Grouping – 2 X 2
m2 – n2 + am + an
[m – n][m + n] + a[m + n]
[m + n] ([m – n] + a)
Look for two pairs of factorable terms – here the first pair are a difference of squares and the second pair have a GCF of “a”
Pull the GCF out in front and then simply write the “left-over” factor from each term.
Factoring each pair gave a binomial that has a GCF glob of “m + n”. Grouping is a premlinary step that MUST lead to a second factoring step. (If your grouping step leads to a dead-end, try re-ordering the terms with different groups. Hint: look for a GCF or PST.)
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Example 13: Factor by Grouping 3 X 1
x2 + 9 – 4y2 + 6x •Can you rearrange the terms to put the three terms of a
PST first followed by the opposite of a perfect square?
•Then rewrite the PST into (glob)2 factored form.
•Now factor this binomial using Difference of Squares
•We will call this “Grouping 3 X 1”
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Example 13: Factor by Grouping 3 X 1x2 + 6x + 9 – 4y2
[x2 + 6x + 9 ] – [4y2]Do you see this as a PST? Isn’t this also a
Can you write it as (glob)2? perfect square?
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Example 12: Factor by Grouping 3 X 1x2 + 6x + 9 – 4y2
[x2 + 6x + 9 ] – [4y2](x + 3) 2 – 4y2
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Example 13: Factor by Grouping 3 X 1
x2 + 6x + 9 – 4y2
[x2 + 6x + 9 ] – [4y2]
(x + 3) 2 – 4y2
[(x + 3) + 2y] [(x + 3) – 2y]
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Examples 14 & 15: Factor by Grouping
a2 – 10a – 49b2 + 25
a2 – 10a + 25 – 49b2
[a2 – 10a + 25 ] – [49b2]
(a – 5) 2 – [49b2]
[(a – 5) + 7b] [(a – 5) – 7b]
ax – ay – bx + by
ax – ay – bx + by
a[x – y ] – b[x – y]
[x – y][a – b]
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Factoring is a basic SKILL for Precalc & Calculus, so PRACTICE
until you are quick & confident!Look for a GCF first and then check for additional steps:
1.Factor GCF for any # terms
2.Difference of Squares binomials
3.Sum or Difference of Cubes binomials
4.PST (Perfect Square Trinomial) trinomials
5.Reverse of FOIL trinomials
6.Factor by Grouping usually for 4 or more terms
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Factor each expression completely.Bring your written work and questions
with you to class tomorrow!
1. 4a3b – 36ab3
2. 2x4y – 12xy – 54y
3. 4x2 – 20x + 25 – 100y2
4. 3a + 3b – 5ac – 5bc
5. 80m3n – 270n4