04/20/23 Math 120 - KM 1

Chapter 6:Introduction to

Polynomials and Polynomial Functions

• 6.1 Introduction to Factoring

• 6.2 FactoringTrinomials: x2+bx+c

• 6.3 FactoringTrinomials: ax2+bx+c

• 6.4 Special Factoring

• 6.5 Factoring: A General Strategy

• 6.6 Applications

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6.1 Introduction to Factoring

6.1

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Let’s Build theGreatest Common Factor

of 90x2y3z and 50y4z5

The GCF of 90x2y3z and 50y4z5 is the product of the “common” bases raised to the smallest exponent.

or

zyx90 32

54zy50

zyx532 322

542 zy52

zy52 3 zy10 3

6.1

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Let’s Build theGreatest Common Factor

of 21x2z and 10y4

21x2z and 10y4 have no common factors!

The only factor common to both expressions is 1.

21x2z and 10y4 are RELATIVELY PRIME

because their GCF is 1.

zx21 2

4y10

zx73 2

4y52

6.1

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Factoring out the GCFis Reversing

the Distributive Property

6.1

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Factor out the GCFfrom 12x5 + 20x3

12x5 20x3

4x3 4x33x2 5

4x3 3x2 5

6.1

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Factor: 12x5 + 20x3

12x5 + 20x3= 4x3(3x2 + 5)

6.1

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Factor: 6x2y5 - 8x3y4

6x2y5 - 8x3y4= 2x2y4(3y - 4x)

6.1

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Factor:9x3 – 11y2 + 3

9x3 – 11y2 + 3

6.1

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Factoring

6.1

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Factor a Tricky One!

x(x + 2) – 6(x + 2)

x(x + 2) – 6(x + 2)

= ( x + 2 )( x – 6 )

6.1

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Another Tricky One!

(x - 7)3x +(x - 7)5

(x-7)3x + (x-7)5

= ( x - 7 )( 3x + 5 )

6.1

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Factor by Grouping:Example 1:

REVERSE FOIL

ab + 7b – 3a – 21

= b(a + 7)– 3(a + 7)

= (a + 7)(b - 3)

(a + 7)(b – 3) = ab – 3a + 7b - 21

6.1

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Factor by Grouping:Example 2:

REVERSE FOIL

x2 + 3x + 5x + 15

= x(x + 3)+ 5(x + 3)

= (x + 3)(x + 5)

6.1

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Factor by Grouping:Example 3:

REVERSE FOIL

x2 + 5x – 5x - 25

= x(x + 5)- 5(x + 5)

= (x + 5)(x - 5)

6.1

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Factor by Grouping:Example 4:

REVERSE FOIL

x2 - 9x + 11x - 99

= x(x - 9)+ 11(x - 9)

= (x + 11)(x - 9)

6.1

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Factor by Grouping:Example 5:

REVERSE FOIL

x3 - 10x2 - 10x + 100

= x2(x - 10)- 10(x - 10)

= (x2 – 10)(x - 10)

6.1

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Factor by Grouping:Example 6:

REVERSE FOIL

18x2 - 21x + 30x - 35

= 3x(6x - 7)+ 5(6x - 7)

= (6x - 7)(3x + 5)

6.1

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Factor by Grouping:Example 7:

REVERSE FOIL

25x2 + 35x + 35x + 49

= 5x(5x + 7)+ 5(5x + 7)

= (5x + 7)(5x + 7)

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Where We Left Off Last Class

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4.4 & 4.5 FactoringTrinomials:

ax2+bx+c

6.2

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First, Let’s ReviewFactor by Grouping

ab + 7b – 3a – 21

x2 + 2x + 10x + 20

6.2

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Now, Let’s Review FOIL!

1512108 2 xxx

)x)(x( 5432

1528 2 xx

Aha! FL = OI (8)(-15) = (10)(-

12)-120 = - 120

6.2

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What’s the Diamond?

ax2 + bx + c

Add tob

Multiply toac

6.2

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2x2 - 11x -40

Add to-11

Multiply to-80

1 80

2 40

3 ---

4 20

5 16

6 ---

7 ---

8 10980 6.2

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2x2 - 11x -40

Add to-11

Multiply to-80

)x()x(x 52852 401652 2 xxx

40112 2 xx

)x)(x( 852 6.2

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6x2 - 17x +12

Add to-17

Multiply to72

)x()x(x 433432 12986 2 xxx

12176 2 xx

)x)(x( 3243

1 72

2 36

3 24

4 18

5 --

6 12

7 --

8 9

872 6.2

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Start with the GCF

)x()x(x 5352

)xxx( 15352 2

)xx( 1522 2

3042 2 xx

)x)(x( 352 6.2

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More Problems?

12y3 + 22y2 – 70y

+ 15x - 4x2 - 9

5ax3 + 20ax2 – 160ax

2x4 + 5x2 + 12

2x6 + 4x3 – 306.2

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6.3 Special Factoring

6.3

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Special FactoringShortcuts

6.3

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Special Polynomials

6.3

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Perfect Trinomial Square

22 2 yxyx )yx)(yx(

2)yx( 6.3

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Perfect Trinomial Square

49142 xx)x)(x( 77

27)x( 6.3

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Perfect Trinomial Square

25102 yy)y)(y( 55

25)y( 6.3

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Perfect Trinomial Square

22 25309 yxyx

)yx)(yx( 5353

253 )yx(

6.3

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OK – Short Cut Time!

498436 2 xx276 )x(

259081 2 xx259 )x(

11664 2 xx218 )x(

6.3

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Difference of Squares

22 yx )yx)(yx(

6.3

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You can do this!

259 2 x)x)(x( 5353

6.3

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Check these out!

4936 2 x)x)(x( 7676

2581 2 xsquaresofsum

116 4 x)x)(x)(x( 121214

6.3

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Sum or Difference of Cubes

n n cubed

1 1

2 8

3 27

4 64

5 125

6 216

… …

n n36.3

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Sum or Difference of Cubes

33 yx )yxyx)(yx( 22

33 yx

)yxyx)(yx( 22 6.3

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Sum or Difference of Cubes

8125 3 x

)xx)(x( 4102525 2

2x5

6.3

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Sum or Difference of Cubes

6427 3 x

)xx)(x( 1612943 2

4x3

6.3

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How about a harder one?

33 343216 yx

)yxyx)(yx( 22 49423676

y7x6

6.3

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6.4 Factoring: A General Strategy

6.4

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Factoring StrategyGCF

1) GREATEST COMMON FACTOR

Check carefully to see if there is a GCF and factor it

out.

If the leading coefficient is negative, factor out -1.

6.4

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Factoring StrategyNumber of Terms

2) Number of TERMS

a) Four Terms: Try grouping

b) Three Terms:

i) a2 + 2ab + b2 Perfect Square ii) a2 – 2ab + b2 Perfect Square iii) ax2 + bx + c UNFOIL c) Two Terms:

i) a2 - b2 Difference of Squares ii) a2 + b2 Sum of Squares - NF iii) x3 – y3 Difference of Cubes iv) x3 + y3 Sum of Cubes

6.4

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Factor Completely: Example 1

2x3 + 6x2 – 8x - 24

= 2[ x3 + 3x2 – 4x – 12 ]

= 2[ x2(x + 3) – 4(x+3) ]

= 2[(x + 3)(x2 – 4)]

= 2[(x + 3)(x + 2)(x - 2)]

= 2(x + 3)(x + 2)(x - 2)

6.4

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Factor Completely: Example 2

5x3 - 80x2 + 320x

= 5x[ x2 – 16x + 64 ]

= 5x[(x - 8)(x - 8)]

= 5x(x - 8)2

6.4

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Factor Completely: Example 3

9x2 + 12x - 5

12

-45-3 15

= 9x2 -3x + 15x - 5

= 3x(3x – 1) + 5(3x - 1)

= (3x – 1)(3x + 5)

6.4

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Factor Completely: Example 4

125x3 + 8y3

= (5x + 2y)(25x2 – 10xy + 4y2)

= (5x)3 + (2y)3

6.4

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Factor Completely: Example 5

x2 + 10x – y2 + 25

= x2 + 10x + 25 – y2

= (x + 5)2 – y2

= [(x + 5) + y] [(x + 5) - y]

= (x + 5 + y)(x + 5 – y)

6.4

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4.8 Applications

6.4

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General Strategy forSolving Equations UsingThe Zero Factor Property

1) Arrange the equation so that one side is zero.

2) Completely factor the other side.

3) Set each factor equal to zero and solve, if possible.

4) Write the solution set.

5) Check each solution by substitution.

6.5

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Zero Factor PropertySolve: 2x(x + 5)(x-3) =

0

0352 xxx

02 0x

350 ,,

05 x5x

03 x3x

6.5

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Zero Factor PropertySolve: (2x - 7)(4x + 3)=

0

03472 xx

4

3

2

7,

072 x72 x

034 x34 x

2

7x

4

3x

6.5

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Zero Factor PropertySolve: 6x2 = 3x

x3x6 2

21

,0

03 0x 1x2

21

x

Use the properties of equality to rearrange the terms of the equation

so that it is equal to ZERO.

0x3x6 2 01x2x3

012 x

6.5

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Solve: x2 = 169

01692 x

01313 )x)(x(

013 x 013 xor

13x 13xor

1313,

6.5

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Solve: x2 + 25 = 10x

025102 xx

055 )x)(x(

05 x 05 xor

5x 5xor

5

6.5

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Solve: 3x2 = 2 - x

023 2 xx

0123 )x)(x(

023 x 01xor

3

2x 1xor

13

2,

6.5

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Solve: 2x3 + 3x2 = 18x + 27

0271832 23 xxx

03292 )x)(x(

03 x 03 x

or3x 3xor

0329322 )x()x(x

03233 )x)(x)(x(

032 x

2

3x

2

333 ,,

6.5

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The Pool is Cool!

Pat has a rectangular swimming pool. The

length is 16 feet longer than the width. The

surface area of the pool is 420 square feet.

What are the dimensions of the pool?

6.5

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Let’s see a Diagram!

w

w + 16

Area = length x width

420 = (w+16)(w)w2 + 16w – 420

= 0(w - 14)(w + 30) = 0w = 14 or w = -306.5

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Answer the Question!

w

w + 16

Pat’s pool is 14 feet wide and 30 feet long.

= 14 feet

= 14 + 16 = 30 feet

6.5

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Is it “Square”?

Lilly and Mike are building a deck and want to make

sure it is “square” (the corners are 90 degrees). If the deck is 12’ by 16’,

what diagonal measurement is needed to

be sure it is “square”?

12’

16’

d

6.5

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Time for the Pythagorean

Equation!

222 cba 222 1612 d)()(

2256144 d2400 d

12’

16’

d

6.5

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Solve for d

04002 d

02020 )d)(d(

2400 d

d = -20 or d = 20If the diagonal is 20’ long, the deck will be

“square”.6.5

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That’s All For Now!