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: GCF FIRST STEP for every expression: factor the GCF! 5x 3 – 10x 2 – 5x

Example 1: GCF FIRST STEP for every expression: factor the GCF! 5x 3 – 10x 2 – 5x 5x(x 2 – 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: 1 st term 2 nd 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: (x – 1)

Example 3: Difference of Squares 75x 4 – 108y 2 GCF first! 3(25x 4 – 36y 2 ) 3(5x 2 – 6y) (5x 2 + 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! a 2 + b 2  PRIME a 2 – b 2  (a + b)(a – b)

Example 4: Sum/Difference of Cubes a 3 - b 3

( ) Example 4: Sum/Difference of Cubes

a 3 - b 3 (a - b) ( ) Cube roots w/ original sign in the middle Example 4: Sum/Difference of Cubes

a 3 - b 3 (a - b) (a 2 + b 2 ) Squares of those cube roots. Note that squares will always be positive.

Example 4: Sum/Difference of Cubes a 3 - b 3 (a - b) (a 2 + ab + b 2 ) The opposite of the product of the cube roots

p (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 (p )

(4x 2 + 9y 2 ) 8x y 3 (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 m 6 – 125n 3 (m 2 – 5n) Cube roots of each Squares of those cube roots & with same sign opp of product of roots in middle (m n 2 )+ 5m 2 n

Example 7: Special Case * 1 st step: Diff of Squares * 2 nd step: Sum/Diff of Cubes x 6 – 64y 6 ( ) ( )( )

Example 7: Special Case * 1 st step: Diff of Squares * 2 nd step: Sum/Diff of Cubes x 6 – 64y 6 (x 3 – 8y 3 ) (x 3 + 8y 3 ) ( )( )

Example 7: Special Case * 1 st step: Diff of Squares * 2 nd step: Sum/Diff of Cubes x 6 – 64y 6 (x 3 – 8y 3 ) (x 3 + 8y 3 ) (x–2y)(x 2 +2xy+4y 2 ) (x+2y)(x 2 -2xy+4y 2 )

Example 8: PST 9x 2 – 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 9x 2 – 30x + 25 (3x – 5) 2

Example 9: Reverse FOIL (Trial & Error) 6x 2 – 17x + 12 ( ) 3x – 42x – 3

Reverse FOIL (Trial & Error) Hint: don’t forget to read the “signs” ax 2 + bx + c  ( + )( + ) ax 2 – bx + c  ( – )( – ) ax 2 + bx – c  ( + )( – ) positive product has larger value ax 2 – bx – c  ( + )( – ) negative product has larger value

Special Case 10: Some quartics can be factored like quadratics (x 4  x 2 ● x 2 ) x 4 – 5x 2 – 36 ( ) x 2 – 9x But, you aren’t done yet! Do you see why? (x + 3)(x – 3)(x 2 + 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) x 3 – 2x 2 + 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 x 3 – 2x 2 + ax – 2a x 2 [x – 2 ] + a [x – 2] [x – 2 ] Glob is a GCF “Left-Over”Factors (x 2 + a)

Summary: Factor by Grouping 2 X 2 x 3 – 2x 2 + ax – 2a Look for two small [x 3 – 2x 2 ] +[ ax – 2a] factorable groups! x 2 [x – 2] + a [x – 2] Check IF same leftover factor (glob)! [x – 2 ] (x 2 + a) Pull the final GCF out in front of the leftover factors.

Example 12: Factor by Grouping – 2 X 2 m 2 – n 2 + 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 m 2 – n 2 + 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 x – 4y 2 + 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 1 x 2 + 6x + 9 – 4y 2 [x 2 + 6x + 9 ] – [4y 2 ] 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 1 x 2 + 6x + 9 – 4y 2 [x 2 + 6x + 9 ] – [4y 2 ] (x + 3) 2 – 4y 2

Example 13: Factor by Grouping 3 X 1 x 2 + 6x + 9 – 4y 2 [x 2 + 6x + 9 ] – [4y 2 ] (x + 3) 2 – 4y 2 [(x + 3) + 2y] [(x + 3) – 2y]

Examples 14 & 15: Factor by Grouping a 2 – 10a – 49b a 2 – 10a + 25 – 49b 2 [a 2 – 10a + 25 ] – [49b 2 ] (a – 5) 2 – [49b 2 ] [(a – 5) + 7b] [(a – 5) – 7b] 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! 1.4a 3 b – 36ab 3 2.2x 4 y – 12xy – 54y 3.4x 2 – 20x + 25 – 100y 2 4.3a + 3b – 5ac – 5bc 5.80m 3 n – 270n 4