W A T K I N S - J O H N S O N C O M P A N Y Semiconductor Equipment Group

1
Chabot Mathematics
5.5 FactorSpecial Forms
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu
2
Review

• Any QUESTIONS About
• 5.4 ? Factoring TriNomials
• Any QUESTIONS About HomeWork
• 5.4 ? HW-18

3
5.5 Factoring Special Forms

• Factoring Perfect-Square Trinomials and
  Differences of Squares
• Recognizing Perfect-Square Trinomials
• Factoring Perfect-Square Trinomials
• Recognizing Differences of Squares
• Factoring Differences of Squares
• Factoring SUM of Two Cubes
• Factoring DIFFERENCE of Two Cubes

4
Recognizing Perfect-Sq Trinoms

• A trinomial that is the square of a binomial is
  called a perfect-square trinomial
• A2 2AB B2 (A B)2
• A2 - 2AB B2 (A - B)2
• Reading the right sides first, we see that these
  equations can be used to factor perfect-square
  trinomials.
• A2 2AB B2 (A B)(A B)
• A2 - 2AB B2 (A - B)(A - B)

5
Recognizing Perfect-Sq Trinoms

• Note that in order for the trinomial to be the
  square of a binomial, it must have the
  following
• 1. Two terms, A2 and B2, must be squares, such
  as 9, x2, 100y2, 25w2
• 2. Neither A2 or B2 is being SUBTRACTED.
• 3. The remaining term is either 2 ? A ? B or
  -2 ? A ? B
• where A B are the square roots of A2 B2

6
Example ? Trinom Sqs

• Determine whether each of the following is a
  perfect-square trinomial.
• a) x2 8x 16 b) t2 - 9t - 36
• c) 25x2 4 20x
• SOLUTION a) x2 8x 16
• Two terms, x2 and 16, are squares.
• Neither x2 or 16 is being subtracted.
• The remaining term, 8x, is 2?x?4, where x and 4
  are the square roots of x2 and 16

7
Example ? Trinom Sqs

• SOLUTION b) t2 9t 36
• Two terms, t2 and 36, are squares.
• But 36 is being subtracted so t2 9t 36 is not
  a perfect-square trinomial
• SOLUTION c) 25x2 4 20x
• It helps to write it in descending order.
• 25x2 20x 4

8
Example ? Trinom Sqs

• SOLUTION c) 25x2 - 20x 4
• Two terms, 25x2 and 4, are squares.
• There is no minus sign before 25x2 or 4.
• Twice the product of the square roots is 2 ? 5x ?
  2, is 20x, the opposite of the remaining term,
  -20x
• Thus 25x2 - 20x 4 is a perfect-square
  trinomial.

9
Factoring a Perfect-Square Trinomial

• The Two Types of Perfect-Squares
• A2 2AB B2 (A B)2
• A2 - 2AB B2 (A - B)2

10
Example ? Factor Perf. Sqs

• Factor a) x2 8x 16
• b) 25x2 - 20x 4
• SOLUTION a)
• x2 8x 16 x2 2 ? x ? 4 42 (x 4)2
• A2 2 A B B2 (A B)2

11
Example ? Factor Perf. Sqs

• Factor a) x2 8x 16
• b) 25x2 - 20x 4
• SOLUTION b)
• 25x2 20x 4 (5x)2 2 ? 5x ? 2 22 (5x
  2)2
• A2 2 A B B2 (A B)2

12
Example ? Factor 16a2 24ab 9b2

• SOLUTION
• 16a2 - 24ab 9b2 (4a)2 - 2(4a)(3b) (3b)2
• (4a - 3b)2 (4a - 3b)(4a - 3b)
• CHECK
• (4a - 3b)(4a - 3b) 16a2 - 24ab 9b2 ?
• The factorization is (4a - 3b)2.

13
Expl ? Factor 12a3 108a2 243a

• SOLUTION
• Always look for a common factor. This time there
  is one. Factor out 3a.
• 12a3 - 108a2 243a 3a(4a2 - 36a 81)
• 3a(2a)2 - 2(2a)(9) 92
• 3a(2a - 9)2
• The factorization is 3a(2a - 9)2

14
Recognizing Differences of Squares

• An expression, like 25x2 - 36, that can be
  written in the form A2 - B2 is called a
  difference of squares.
• Note that for a binomial to be a difference of
  squares, it must have the following.
• There must be two expressions, both squares, such
  as 9, x2, 100y2, 36y8
• The terms in the binomial must have different
  signs.

15
Difference of 2-Squares

• Diff of 2 Sqs ? A2 - B2
• Note that in order for a term to be a square, its
  coefficient must be a perfect square and the
  power(s) of the variable(s) must be even.
• For Example 25x4 - 36
• 25 52
• The Power on x is even at 4 ? x4 (x2)2
• Also, in this case 36 62

16
Example ? Test Diff of 2Sqs

• Determine whether each of the following is a
  difference of squares.
• a) 16x2 - 25 b) 36 - y5 c) -x12 49
• SOLUTION a) 16x2 - 25
• The 1st expression is a sq 16x2 (4x)2
• The 2nd expression is a sq 25 52
• The terms have different signs.
• Thus, 16x2 - 25 is a difference of squares, (4x)2
  - 52

17
Example ? Test Diff of 2Sqs

• SOLUTION b) 36 - y5
• The expression y5 is not a square.
• Thus, 36 - y5 is not a diff of squares
• SOLUTION c) -x12 49
• The expressions x12 and 49 are squaresx12
  (x6)2 and 49 72
• The terms have different signs.
• Thus, -x12 49 is a diff of sqs, 72 - (x6)2

18
Factoring Diff of 2 Squares

• A2 - B2 (A B)(A - B)

• The Gray Area by Square Subtraction

• The Gray Area by(LENGTH)(WIDTH)

19
Example ? Factor Diff of Sqs

• Factor a) x2 - 9 b) y2 - 16w2
• SOLUTION a) x2 - 9 x2 32 (x 3)(x - 3)
• A2 - B2 (A B)(A - B)

b) y2 - 16w2 y2 - (4w)2 (y 4w)(y - 4w)
A2 - B2 (A B) (A - B)
20
Example ? Factor Diff of Sqs

• Factor c) 25 - 36a12 d) 98x2 - 8x8
• SOLUTION
• c) 25 - 36a12 52 - (6a6)2 (5 6a6)(5 - 6a6)
• d) 98x2 - 8x8
• Always look for a common factor. This time
  there is one, 2x2
• 98x2 - 8x8 2x2(49 - 4x6)
• 2x2(72 - (2x3)2
• 2x2(7 2x3)(7 - 2x3)

21
Grouping to Expose Diff of Sqs

• Sometimes a Clever Grouping will reveal a
  Perfect-Sq TriNomial next to another Squared Term
• Example ?Factor m2 - 4b4 14m 49
• ? rearranging ?
• m2 14m 49 - 4b4
• ? GROUPING ?
• (m2 14m 49) - 4b4

22
Grouping to Expose Diff of Sqs

• Example ?Factor m2 - 4b4 14m 49
• Recognize m2 14m 49 as Perfect Square
  Trinomial ? (m7)2
• Also Recognize 4b4 as a Sq ? (2b)2
• (m2 14m 49) - 4b4
• ? Perfect Sqs ?
• (m 7)2 - (2b2)2
• In Diff-of-Sqs Formula A?m7 B?2b2

23
Grouping to Expose Diff of Sqs

• Example ?Factor m2 - 4b4 14m 49
• (m 7)2 - (2b2)2
• ? Diff-of-Sqs ? (A - B)(A B) ?
• (m7 - 2b2)(m 7 2b2)
• ? Simplify ? ReArrange ?
• (-2b2 m 7)(2b2 m 7)
• The Check is Left for us to do Later

24
Factoring Two Cubes

• The principle of patterns applies to the sum and
  difference of two CUBES. Those patterns
• SUM of Cubes

• DIFFERENCE of Cubes

25
TwoCubes SIGN Significance

• Carefully note the Sum/Diff of Two-Cubes Sign
  Pattern

SAME Sign
OPP Sign
SAME Sign
OPP Sign
26
Example Factor x3 64

• Factor

Recognize Pattern as Sum of CUBES
Determine Values that were CUBED
Map Values to Formula
Substitute into Formula
Simplify and CleanUp
27
Example Factor 8w3-27z3

• Factor

Recognize Pattern as Difference of CUBES
Determine CUBED Values
Simplify by Properties of Exponents
Map Values to Formula
Sub into Formula
Simplify CleanUp
28
Example Check 8w3-27z3

• Check

Use Distributive property
Use Comm Assoc. properties, and Adding-to-Zero
?
29
Sum Difference Summary

• Difference of Two SQUARES

• SUM of Two CUBES

• Difference of Two CUBES

30
Factoring Completely

• Sometimes, a complete factorization requires two
  or more steps. Factoring is complete when no
  factor can be factored further.
• Example Factor 5x4 - 3125
• May have the Difference-of-2sqs TWICE

31
Factoring Completely

• SOLUTION
• 5x4 - 3125 5(x4 - 625)
• 5(x2)2 - 252
• 5(x2 - 25)(x2 25)
• 5(x - 5)(x 5)(x2 25)
• The factorization 5(x - 5)(x 5)(x2 25)

32
Factoring Tips

• Always look first for a common factor. If there
  is one, factor it out.
• Be alert for perfect-square trinomials and for
  binomials that are differences of squares.
• Once recognized, they can be factored without
  trial and error.
• Always factor completely.
• Check by multiplying.

33
WhiteBoard Work

• Problems From 5.5 Exercise Set
• 14, 22, 48, 74, 94, 110

• The SUM (S) DIFFERENCE (?) of Two Cubes

34
All Done for Today
Sum ofTwoCubes
35
Chabot Mathematics
Appendix
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu

36
Graph y x

• Make T-table

37
(No Transcript)