Factoring special products

1
Lesson 83

• Factoring special products

2
Look for a pattern in the products

• (x1)2 (x1)(x1) x22x1 x2 2(1x)12
• (x2)2 (x2)(x2) x2 4x4 x2 2(2x) 22
• (x3)2(x3)(x3) x26x9 x2 2(3x)32
• (x-1)2(x-1)(x-1) x2 -2x 1 x2
  2(-1x)(-1)2
• (x-2)2 (x-2)(x-2) x2 - 4x 4 x2 2(-2x)
  (-2)2
• Square the first term
• Square the 2nd term
• Multiply the product of both terms by 2

3
Perfect square trinomials

• the factored form of a perfect square trinomial
  is
• a2 2ab b2 (ab)2
• a2-2ab b2 (a-b)2

4
Factoring perfect square trinomials

• Determine if each polynomial is a perfect square
  trinomial. If it is, factor it!
• x2 6x 9
• write in perfect square trinomial form
• x2 2(3x) 32
• (x3)2
• x2 -2x 4
• x2 2(-1x) 22 not perfect square form

5
practice

• 36x2 -48x16
• Factor GCF first
• 4(9x2-12x4)
• Write in perfect square form
• 4 ( (3x)2 2(-3x)(2) 22 )
• 4(3x-2)2

6
Difference of 2 squares

• a2 - b2 (ab)(a-b)
• example
• x2 -49
• x2 -(7)2
• (x-7)(x7)

7
practice

• Are these the difference of 2 squares. If so,
  factor them.
• 4x2 - 25
• 9m2 -16n6
• X2-8
• 100x2 - 25
• 2x6 -288
• X2 9
• -36 x10
• -64z8

8
Lesson 87

• Factoring by grouping

9
Polynomials can be factored by grouping

• When a polynomial has 4 terms, make 2 groups and
  factor out the GCF from each group.
• 2x2 4xy 7x 14y
• (2x2 4xy) (7x 14y)
• 2x(x2y) 7 (x 2y)
• Now factor out the common factor from each
  binomial
• (x2y)(2x7)

10
Factor by grouping

• 5x2 10xy 3x 6y

11
Rearranging before grouping

• Sometimes it is necessary to rearrange the terms
  so that there are common factors
• 3y2 - 8y3 - 8y 3
• Maybe it would be better to rearrange
• 3y2 3 -8y3 -8y
• Then 3(y21) -8y(y21)
• (y21)(3-8y)

12
practice

• 5x2 - 12x3 - 12x 5

13
Factoring with the GCF

• Always factor out GCF first
• 45a3b - 15a3 15a2b - 5a2
• GCF is 5a2
• 5a2(9ab-3a 3b -1)
• Group
• 5a2(9ab-3a) (3b-1)
• 5a2(3a(3b-1) (3b-1))
• 5a2(3b-1)(3a1)

14
practice

• 54xz3 6xz2 - 18z3 - 2z2

15
Factoring with opposites

• factor 3a2b - 18 a 30 -5ab
• Group
• (3a2b-18a) (30-5ab)
• 3a(ab-6) 5(6-ab)
• Factor out -1 from 2nd group
• 3a(ab-6) 5(-1)(-6ab)
• 3a(ab-6) -5(ab-6)
• (ab-6)(3a-5)

16
practice

• 5x2y - 3xy - 50x 30

17
Factoring a trinomial by grouping

• ax2 bx c
• 1) find the product of ac
• 2) find 2 factors of ac with a sum equal to b
• 3) write the trinomial using the sum
• 4) factor by grouping
• Example 2x2 11x 15
• 2 x 15 30
• Find factors of 30 that add up to 11
• 6, 5
• 2x2 5x 6x 15
• Factor by grouping

18
factor

• x2 - 7x - 44
• 6k2 - 17 k 10
• x2 - 4x - 21
• 5k2 - 13k 6

19
Investigation 9

• Choosing a factoring method

20
Checklist for factoring

• 1) look for the GCF- does each term have a common
  factor?
• 2)look for a difference of 2 squares- are there
  only 2 terms of the polynomial and are they
  subtracted?
• 3)look for perfect square trinomials- are the
  first and last terms perfect squares and is the
  second term the product of the square roots of
  the first and last terms?
• 4) are there 3 terms in the polynomial and are
  they all being added-is the last term not a
  perfect square?
• 5)are there 4 terms in the polynomial- if they
  have no GCF can you group them into 2 groups that
  hace common factors?

21
factor

• x2 2x 1
• 3x2 xy - 12x - 4y
• 3x2 13x 4
• x2 9x 20
• 2x2 8x 6
• 5x4 - 5x2
• 9x2 30x 25
• x2 -9