Chabot Mathematics
This presentation is the property of its rightful owner.
Sponsored Links
1 / 37

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected] PowerPoint PPT Presentation


  • 58 Views
  • Uploaded on
  • Presentation posted in: General

Chabot Mathematics. §5.5 Factor Special Forms. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected] MTH 55. 5.4. Review §. Any QUESTIONS About §5.4 → Factoring TriNomials Any QUESTIONS About HomeWork §5.4 → HW-14. §5.5 Factoring Special Forms.

Download Presentation

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Bruce mayer pe licensed electrical mechanical engineer bmayer chabotcollege

Chabot Mathematics

§5.5 FactorSpecial Forms

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]


Review

MTH 55

5.4

Review §

  • Any QUESTIONS About

    • §5.4 → Factoring TriNomials

  • Any QUESTIONS About HomeWork

    • §5.4 → HW-14


5 5 factoring special forms

§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

    • Facting DIFFERENCE of Two Cubes


Recognizing perfect sq trinoms

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)


Recognizing perfect sq trinoms1

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


Example trinom sqs

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


Example trinom sqs1

Example  Trinom Sqs

  • SOLUTION b) t2– 9t– 36

  • Two terms, t2 and 36, are squares. But

  • But 36 is being subtracted so t2– 9t– 36 is nota perfect-square trinomial.

  • SOLUTION c) 25x2 + 4 – 20x

    It helps to write it in descending order.

    25x2– 20x + 4


Example trinom sqs2

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.


Factoring a perfect square trinomial

Factoring a Perfect-Square Trinomial

  • The Two Types of Perfect-Squares

    A2 + 2AB + B2 = (A + B)2

    A2− 2AB + B2 = (A−B)2


Example factor perf sqs

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


Example factor perf sqs1

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


Example factor 16 a 2 24 ab 9 b 2

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.


Expl factor 12 a 3 108 a 2 243 a

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


Recognizing differences of squares

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.


Difference of 2 squares

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


Example test diff of 2sqs

Example  Test Diff of 2Sqs

  • Determine whether each of the following is a difference of squares.

    a) 16x2− 25b) 36 −y5c) −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


Example test diff of 2sqs1

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 squares:x12 = (x6)2 and 49 = 72

  • The terms have different signs.

  • Thus, −x12 + 49 is a diff of sqs, 72− (x6)2


Factoring diff of 2 squares

Factoring Diff of 2 Squares

  • A2−B2 = (A + B)(A−B)

  • The Gray Area by Square Subtraction

  • The Gray Area by(LENGTH)(WIDTH)


Example factor diff of sqs

Example  Factor Diff of Sqs

  • Factor: a) x2− 9b) 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)


Example factor diff of sqs1

Example  Factor Diff of Sqs

  • Factor: c) 25 − 36a12d) 98x2− 8x8

  • SOLUTION

    c) 25 − 36a12 = 52− (6a6)2 = (5 + 6a6)(5 − 6a6)

    d) 98x2− 8x8

    Alwayslook for a common factor. This time there is one, 2x2:

    98x2− 8x8 = 2x2(49 − 4x6)

    = 2x2[(72− (2x3)2]

    = 2x2(7 + 2x3)(7 − 2x3)


Grouping to expose diff of sqs

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


Grouping to expose diff of sqs1

Grouping to Expose Diff of Sqs

  • Example Factor m2− 4b4 + 14m + 49

    • Recognize m2 + 14m + 49 as Perfect Square Trinomial → (m+7)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→m+7; B→2b2


Grouping to expose diff of sqs2

Grouping to Expose Diff of Sqs

  • Example Factor m2− 4b4 + 14m + 49

    (m + 7)2− (2b2)2

     Diff-of-Sqs → (A − B)(A + B) 

    ([m+7] − 2b2)([m + 7] + 2b2)

     Simplify → ReArrange 

    (−2b2 + m + 7)(2b2 + m + 7)

  • The Check is Left for us to do Later


Factoring two cubes

Factoring Two Cubes

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

    • SUM of Cubes

  • DIFFERENCE of Cubes


Twocubes sign significance

TwoCubes SIGN Significance

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

SAME Sign

OPP Sign

SAME Sign

OPP Sign


Example factor x 3 64

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


Example factor 8 w 3 27 z 3

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


Example check 8 w 3 27 z 3

Example: Check 8w3−27z3

  • Check

Use Distributive property

Use Comm & Assoc. properties, and Adding-to-Zero


Sum difference summary

Sum & Difference Summary

  • Difference of Two SQUARES

  • SUM of Two CUBES

  • Difference of Two CUBES


Factoring completely

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


Factoring completely1

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)


Factoring tips

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.


Whiteboard work

WhiteBoard Work

  • Problems From §5.5 Exercise Set

    • 14, 22, 48, 74, 94, 110

  • The SUM (Σ) & DIFFERENCE (Δ) of Two Cubes


All done for today

All Done for Today

Sum ofTwoCubes


Bruce mayer pe licensed electrical mechanical engineer bmayer chabotcollege

Chabot Mathematics

Appendix

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]


Graph y x

Graph y = |x|

  • Make T-table


  • Login