Review of Factoring Methods

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Review of Factoring Methods

• In this slideshow you will see examples of
• Factor GCF ? for any terms
• Difference of Squares ? binomials
• Sum or Difference of Cubes ? binomials
• PST (Perfect Square Trinomial) ? trinomials
• Reverse of FOIL ? trinomials
• Factor by Grouping ? usually for 4 or more terms

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Example 1 GCF

• FIRST STEP for every expression
• factor the GCF!
• 5x3 10x2 5x

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Example 1 GCF

• FIRST 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 (x1) -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 (x1) -2
• 3(x 1)2
• Combine like terms 1 -2

(x 1)
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Example 3Difference 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 4Sum/Difference of Cubes

• a3 - b3

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Example 4Sum/Difference of Cubes

• a3 - b3
• ( ) ( )

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Example 4Sum/Difference of Cubes

• a3 - b3
• (a - b) ( )
• Cube roots w/ original
• sign in the middle

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Example 4Sum/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 4Sum/Difference of Cubes

• a3 - b3
• (a - b) (a2 ab b2)
• The opposite of the product
• of the cube roots

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Example 5Sum/Difference of Cubes

• p3 - 125
• (p - 5)
• Cube roots of each Squares of those cube roots
  
• with same sign opp of product of roots in
  middle

(p2 25)
5p
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Example 5Sum/Difference of Cubes

• 8x3 27y3
• (2x 3y)
• Cube roots of each Squares of those cube roots
  
• with same sign opp of product of roots in
  middle

(4x2 9y2)
6xy
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Example 6Difference 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)
• (x2y)(x22xy4y2) (x2y)(x2-2xy4y2)

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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
• ( ) ( )

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Reverse FOIL(Trial Error)

• Hint dont 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
• ( ) ( )

But, you arent done yet! Do you see why?
Now youre done!
(x 3)(x 3)(x2 4)
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Example 11 Factor by Grouping(4 or more terms)

• a(x 7) b(x 7)
• (x 7) (a b)

Left-over factors
Glob is the GCF

• 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!

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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
(x2 a)
Glob is a GCF
Left-OverFactors
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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
a m n
m n
(m n a)
Glob is a GCF
Left-OverFactors
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Summary Factor by Grouping 2 X 2

• m2 n2 am an

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
m nm n am n
m n (m n a)
Pull the GCF out in front and then simply write
the left-over factor from each term.
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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 1

• x2 6x 9 4y2
• x2 6x 9 4y2
• Do you see this as a PST? Isnt this also a
• Can you write it as (glob)2? perfect
  square?

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Example 12 Factor by Grouping 3 X 1

• x2 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 ax y
bx y x ya 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
• Factor GCF ? for any terms
• Difference of Squares ? binomials
• Sum or Difference of Cubes ? binomials
• PST (Perfect Square Trinomial) ? trinomials
• Reverse of FOIL ? trinomials
• Factor by Grouping ? usually for 4 or more terms

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Factor each expression completely.Bring your
written work and questionswith you to class
tomorrow!

1. 4a3b 36ab3
2. 2x4y 12xy 54y
3. 4x2 20x 25 100y2
4. 3a 3b 5ac 5bc
5. 80m3n 270n4