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

Example 1: GCFFIRST STEP for every expression: factor the GCF!

5x3 – 10x2 – 5x

Example 1: GCFFIRST STEP for every expression: factor the GCF!

5x3 – 10x2 – 5x

5x(x2 – 2x – 1)

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?

Example 2: GCF

3(x + 1)3 – 6(x + 1)2

3(x + 1)2 [ ] this is the GCF

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

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)

Example 3:Difference of Squares

75x4 – 108y2

GCF first! 3(25x4 – 36y2)3(5x2 – 6y) (5x2 + 6y)Recall these binomials are called conjugates.

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)

Example 4:Sum/Difference of Cubes

a3 - b3

a3 - b3

( ) ( )

Example 4:Sum/Difference of Cubes

a3 - b3

(a - b) ( )

Cube roots w/ original sign in the middle

Example 4:Sum/Difference of Cubes

Example 4:Sum/Difference of Cubes

a3 - b3

(a - b) (a2 + b2)

Squares of those cube roots.

Note that squares will always be positive.

Example 4:Sum/Difference of Cubes

a3 - b3

(a - b) (a2 + ab + b2)The opposite of the product

of the cube roots

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)

(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

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

Example 7: Special Case * 1ststep: Diff of Squares * 2nd step: Sum/Diff of Cubes

x6 – 64y6

( ) ( ) ( )( ) ( )( )

Example 7: Special Case * 1ststep: Diff of Squares

* 2nd step: Sum/Diff of Cubes

x6 – 64y6

(x3 – 8y3) (x3 + 8y3)

( )( ) ( )( )

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)

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?

Example 8: PST

9x2 – 30x + 25

(3x – 5) 2

Example 9: Reverse FOIL (Trial & Error)

6x2 – 17x + 12

( ) ( ) 3x – 4 2x – 3

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

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!

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

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”

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)

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 .

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]

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.)

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”

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?

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]

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]

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

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