Beginning Algebra Professor Sikora MAT0024C

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Beginning Algebra Professor SikoraMAT0024C
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FACTORING and QUADRATIC EQUATIONS
CHAPTER 5
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5.1 Factors G. C. Fs

• In multiplying given factors ?find the product
  ex 2 5 7 70
• In factoring given product ?find the factors
  ex 70 2 5 7
• A w/ exactly 2 factors the itself 1
  PRIME number (whole s gt 1)
• Prime Factorization when whole expressed as
  product of prime factors

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5.1 Factors G. C. Fs

• Find the PRIME FACTORIZATION of
• 200
• 2 100
• 50 2
• 10 5
• 2 5
• 23 . 52

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5.1 Factors G. C. Fs

• Monomial in factored form if product of
• 1) Prime s
• 2) Variables (no exponent gt 1) Express in
  factored form
• 45x3y2 77ab2

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5.1 Factors G. C. Fs

• The Greatest Common Factor (G.C.F.) of 2 or more
  integers is the greatest that is a
  factor of all the integers
• 1) Find the Prime Factorization of all the s
• 2) Multiply the common PRIME factors
• Find the G.C.F. of 45, 60, 75
  84 70

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5.1 Factors G. C. Fs

• The Greatest Common Factor (G.C.F.) of 2 or more
  monomials is the product of their common
  factors.
• Find the G.C.F. 20a2b3, 12ab4, 40a3b2 18x3
  6x

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5.1 Factoring Using Distributive Prop.

• A polynomial is in Factored form when expressed
  as a product of monomial polynomial.
• Multiplying Polynomials (Ch. 4)
  x(x 75) x2 75x
• 5x(3x - 4y) 15x2 - 20xy
• Factoring Polynomials (Ch. 5)
• x2 75x x(x 75)
• 15x2 - 20xy 5x(3x - 4y)
• In the above 2 exs., we are factoring out the
  GCF

reverse
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5.1 Factoring out the G.C.F.

• Use the Distributive Prop. to Factor each
  polynomial 1) Find G.C.F. of each term
• 2) Use Distrib. to write product of G.C.F.
  remaining factor Ck by multiplying back!
• Ex 12mn2 - 18m2n2
• Ex 20abc 15a2c - 5ac

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5.1 Factoring out the G.C.F.

• Use the Distributive Prop. to Factor each
• Ex 8p5q2 16p6q3 - 12p4q7
• Ex 2y(y 1) 7(y 1) Common binomial
  factor
• Ex - b4 3b2 2 factor out -1

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5.1 Factoring by Grouping

• Ex. w/ between groups
• 49x 21y2 35x3y 15x2y3
• Ex. w/ - between groups
• 49c - 7cd - 21 3d

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5.1 Factoring by Grouping

• Rules
• 1.) Group terms in 2 groups each w/ common
  factor
• 2.) Factor w/in groups If cant, rearrange
  terms try again
• 3.) Factor out common binomial from entire
  polynomial
• Ex. Need to rearrange terms
• 6y2 20w2 15yw 8yw

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5.1 Factoring by Grouping

• If ALL terms have a GCF, factor it out first
• Ex. -4t 4s 4tz 4sz

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5.2 Factoring Trinomials x2bxc lead coeff.
1

• Review Factor . Factor Product
  Binomial . Binomial Trinomial FOIL
• FOIL (x 2)(x 3) ________________
• Reverse FOIL Product Factor . Factor
• Given x2 5x 6 Find 2 s that mult. to 6
  add to 5
• ADD MULTIPLY ans. 2 3
• So x2 5x 6 (x 2)(x 3)

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5.2 Factoring Trinomials lead coeff. 1

• Ex Factor x2 12x 20 Prod. 20
  Sum 12 s?
• Ans (x __)(x __) Ck by FOILing
• Ex Factor x2 - 9x 22
• Ex Factor x2 - 2x 5 Its Prime!

_______ ________

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5.2 Factoring Trinomials lead coeff. 1

• Facts in Factoring x2 bx c
• Find 2 integers whose prod. c and sum b
• Both integers positive if b c both positive,
• Both ints. negative if c positive and b neg.
• 1 integer positive 1 neg. if c neg.
• Ex Factor q2 2q - 24
• Ex Factoring out common monomial 1st
• Factor 3x4 - 15x3 18x2

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5.2 Factoring Trinomials lead coeff. 1

• Trinomials in 2 variables factor best if both
  variables are placed in ( )( )
  1st
• Ex Factor r2 6rs 8s2 ( )(
  )
• Ex Factor s2 6st 7t2 ( )(
  )

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5.3 Factoring Trinomials ax2bxc lead coeff. ?
1

• Trial Check Factoring Method
• Factor 3x2 7x 2 ( )(
  )
• FOIL CK!
• Factor 6h2 7hk 2k2 ( )(
  )
• FOIL CK!

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5.3Factoring Trinomials lead coeff. ? 1 Group
Method

• Review Factor . Factor Product
  Binomial . Binomial Trinomial FOIL
• FOIL (3x 2)(5x 3) ________________
• PRODUCT of 1st Last term of trinomial _____
• Find all double factors of this 90 say yes to
  the pair that SUM up to middle term

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5.3 Group Factoring Trinomials lead coeff. ? 1

• To Factor Trinomial 1) Find factors of (1st
  term . Last term product)
• 2) Which factor pair sums up to middle term yes
• 3) Rewrite the trinomial w/ yes factors added
  in parentheses as middle coefficient
• 4) Factor by grouping
• 5)FOIL check ? Ex 2x2 9x 10
  
• Factors of 20 gt20 . 1 N
• 10 . 2 N 5 . 4 Y

Prod. 20
Sum
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5.3 Group Factoring Trinomials lead coeff. ? 1

• Ex continued 2x2 9x 10 Prod. 20 5 .
  4Y Sum 9
• 3)Rewrite the trinomial w/ yes factors added in
  parentheses as middle coefficient
• 4)Factor by grouping 2x2 5x 4x 10
• x(2x 5) 2(2x 5)
• 5)FOIL check ? (2x 5)(x 2)

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5.3 Group Factoring Trinomials lead coeff. ? 1

• Follow 5 steps for Factoring Trinomials
• Ex 3a2 13a 4 Ex 6x2 25x 14

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5.3 Group Factoring Trinomials lead coeff. ? 1

• Follow 5 steps for Factoring Trinomials
• Ex 3p2 4p 1 Ex 6x2 19x 10

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5.3 Group Factoring Trinomials lead coeff. ? 1

• Follow 5 steps for Factoring Trinomials
• Ex 5x2 13x 3 Ex 21a2 13a 2

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5.3 Group Factoring Trinomials lead coeff. ? 1

• Follow 5 steps for Factoring Trinomials
• Ex 2 variables Ex Common Factor
• 6m2 11mn 10n2 12a2 22a - 20

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Mini-Quiz 5.1?5.3 1-2) 12a3b 15ab4

• 1) GCF ____ 2) Factor ___(____________)
• 3-4) a3x3 - a2x2 2ax 3) GCF ____
  4) Factor
  ___(_________________)
• 5-6) Factor by Grouping 5) 3a2 - 2b - 6a ab
  6) w3 - w2 w - 1
• 7-10) Factor
• 7) m2 10m 21 8) 5b2 20b - 60
• 9) 5n2 7n 2 10) 2x2 - 5x - 3

GCF?
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5.4 Factoring - Perfect Squares

• Find the Area of this square 2 different ways
• x y
• x
• y
• Rule (x y)2 x2 2xy y2
• Rule (x - y)2 x2 2xy y2

PERFECT SQUARE TRINOMIAL
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5.4 Factoring - Perfect Squares

• Determine if each is a PERFECT TRINOMIAL, if so
  FactorCaution 1st arrange descending
• x2 - 12x 36 Y N
  (__________)2
• x2 14x - 49 Y N
  (__________)2
• 4y2 36yz 81z2 Y N (__________)2
• 9n2 49 - 21n
• ______________ Y N (__________)2

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5.4 Factoring Differences of Squares

• FOIL Chapter 4 product of (x 8)(x - 8)
• x2 8x - 8x - 64 x2 - 64 Diff. of 2 sqs x
  64
• Rule x2 - y2 (x y)(x - y)
• Ex m2 - 81 Ex 100x2 - 25y2
• m2 - 92 GCF ___
• ( )( ) ___( )
• ___( )2 - ( )2
• ___( )( )

Can binomial can be written as x2 - y2 ?
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5.4 Factoring Differences of Squares

• Rule x2 - y2 (x y)(x - y) Look for GCF!
• Ex 16a2 - 25b2 Ex 20cd2 - 125c5

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5.4 Factoring Differences of Squares

• Rule x2 - y2 (x y)(x - y) Keep going!
• Ex 3k4 - 48 Ex 16x4 - y4

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5.6 Start Factor Flow Chart

Y
Y
4
GCF?
of terms
2
GCF ?
Factor it out
Factor it out
N
N
3
Grouping ( )( )
Diff of squares (x y)(x - y)
Y
GCF?
Factor it out
Prod/ Sum/ Group ( )( )
N
Perfect Trinomial?
N
Leading Coeff. 1?
N
Y
Y
(x y)2 or (x - y)2
Reverse Foil( )( )
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5.6 Factor via the Flow Chart

• Use the flow chart to decide how to factor each
• 4k2 - 100
• 10x2 - x - 21
• 6p4q 3p3q -12p2q2 - 6pq2

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5.7 Solve Quadratic Eqs. by Factoring

• 1st Degree polynomials in Linear Equations
  Ex 6x 12 0 Solve ________ 1
  solution
• 2nd Degree polynomials in Quadratic Equations
  Ex 6x2 3x 0 2 solutions
• Quadratic Equation in form ax2 bx c 0
  (a ? 0) a, b, c ? Reals
• Quadratic Form
• If equation isnt in Quadratic Form, use -
  properties to get 0 alone on rt. side

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5.7 Solve Quadratic Eqs. by Factoring

• ZERO PRODUCT PROPERTY For all s a b, IF
  a . b 0
• THEN a 0, b 0, or both a and b 0
• Ex (x 3)(x - 5) 0 Solve Check
• Ex (2a 4)(a 7) 0

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5.7 Solve Quadratic Eqs. by Factoring

• ZERO PRODUCT PROPERTY IF a . b 0
• THEN a 0, b 0, or both a and b 0
• Ex x2 9x Dont ? both sides by a variable
  cuz it could 0
• Ex x2 - 36 5x
• Ex a2 - 24a -144

Solve check
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5.7 Solve Quadratic Eqs. by Factoring

• Ex 4m2 25 20m
• Ex x3 2x2 15x Look for common factor
• Ex a3 - 13a2 42a 0 Look for common factor

Solve check
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5.7 Applications of Quadratic Equations

• Ex The length of a hall is 5 times the width.
  The area of the floor is 45 sq. meters. Find the
  halls length width.
• Mark diagram with the facts

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5.7 Applications of Quadratic Equations

• Ex The width of a rectangular yard is 4 meters
  less than the length. The area is 92 more than
  the perimeter. Find the length width.
• Mark diagram with the facts

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5.7 Applications of Quadratic Equations

• Pythagorean Formula
• a2 b2 c2
• Ex The hypotenuse of a rt. ? is 3 longer than
  the longer leg. The shorter leg is 3 shorter
  than the longer leg. Find the lengths of all
  sides.
• Mark diagram with the facts

c
a
b
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Mini-Quiz 5.4?5.7 1-3) Perf. Sq. Trinomials?

• If Y, FACTOR If N, why? 1) x2 10x 25 Y N
  (_____)2
• 2) w2 6w - 9 Y N (_____)2 3) 25y2
  30yz 9z2
• Y N (_____)2
• 4-6) Factor 4) 5a2x3y 20b2xy
• 5) 2n4p 32p 6) 81p4 16q4 7 9) Solve by
  Factoring 7) x2 2x - 15 0
• 8) Check 7 All Solutions!
• 9) k3 8k2 - 12k 10) w The length of
  the blue rectangle is 1 cm more than twice the
  width. The Area 36 sq. cm. Find the Perimeter.
  Solve by factoring!