1 Exponent Rules and Monomials Standards 3 and 4 Simplifying Monomials: Problems POLYNOMIALS Monomials and Polynomials: Adding and Multiplying Multiplying

1

Exponent Rules and Monomials

Standards 3 and 4

Simplifying Monomials: Problems

POLYNOMIALS

Monomials and Polynomials: Adding and Multiplying

Multiplying Binomials: FOIL with MODELING

Multiplying Polynomials with MODELING

Dividing Polynomials: Long Division

Synthetic Division of Polynomials

Greatest Common Factor: GCF

Factoring Polynomials: 2 Terms with MODELING

Factoring Polynomials: Perfect Square Trinomials with MODELING

Factoring Polynomials: General

END SHOW

PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

http://www.mrperezonlinemathtutor.com/

2

STANDARD 3:

Students are adept at operations on polynomials, including long division.

STANDARD 4:

Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes.

ALGEBRA II STANDARDS THIS LESSON AIMS:



3

ESTÁNDAR 3:

Los estudiantes son capaces de hacer operaciones de polinomios, incluyendo division larga.

ESTÁNDAR 4:

Los estudiantes factorizan diferencias de cuadrados, trinomios cuadrados perfectos, y la suma y diferencia de dos cubos.



4

Standards 3 and 4MONOMIALS

Negative Exponents:

a =-nn

1

a= a-n

n1

aand

For any real number a, and any integer n, where a = 0

21

a 61

xa =-2 x =-6 y =-8

81

y

= z-3 = b-73

1

z 71

b



5


am an = a m+n

Multiplying Powers:

For any real number a and integers m and n

= x3+5 = x 8x x 3 5

y y y 2 4 7 = y 2+4+7 = y 13

am

an = am-n

Dividing Powers:

For any real number a, except a=0, and integers m and n

= x 8-3 = x 5

=y 9-8

xx

8

3

yy

9

8 = y



6


amn

= amn

Power of a Power:

Suppose m and n are integers and a and b are real numbers. Then the following is true:

= x(4) (3)= x 12x4

3

y57

= y(5) (7)= y 35

Power of a Product:

(ab) n = a bn n

= x y5 5(xy) 5

(-3pr)3= (-3) p r3 3 3 = -27p r3 3

Power of a Quotient:

ab

n= a

b

n

n

ab

-n= b

a

n

nba

n=

=yx

(2)(3)

(3)(3)=

yx

6

9yx

2

3

3

yx

3

2

-5xy

2

3

5

= =xy

(2)(5)

(3)(5) =xy

10

15

Power to the zero:

a0 = 1 (4y) 0

(-3kp)0 = 1

= 1



7

Standards 3 and 4

(4x y )(-2x y z )53 42 2 23 4= (4)(-2)x x y y z 52

2+5 2= -8x y z3+4

= -8x y z7 7 2

-18p r w34 2

36p r w x53 42

18 29 33 31

18 29 33 31

36 2

Finding the GCF between 18 and 36:

22

32

32

18 = 322

36 = 22 32

We take all the numbers that repeat with the least exponent:

2 32GCF= = 18

p r w x4-2 3-3 2-4 -5=-18 .. 18

36.. 18

= p r w x -1 2

-50 -221

p

2w x52

2-=

-18p r w34 2

36p r w x53 42

(4x y )(-2x y z )53 42 2

Simplify the following monomials:



8

Standards 3 and 4

(3k n )(-7k n r )52 26 7 72 2= (3)(-7)k k n n r 56

= -21k n r6+5 72+2

= -21k n r11 4 7

-27a b c57 9

48a b c d 26 83

27 39 33 31

24 21263

48 2

Finding the GCF between 27 and 48:

33

27 = 33

We take all the numbers that repeat with the least exponent:

3GCF=

a b c d7-3 5-6 9-8 -2=-27 .. 3

48.. 3

= a b c d -9 16

-2-1 14

-27a b c57 9

48a b c d26 83

(3k n )(-7k n r )52 26 7

1

22 24

3

48 = 243

a

16bd 2

4-=

c9

Simplify the following monomials:



9

a b c32 4

2 a b c d 56 72

-2

1

= 2 b c d (-3)(-2) (-5)(-2)(-1)(-2) (-3)(-2)

= b c d66 10 4

= b c d66 102 2

Simplify the following monomial:

= a b c d -5-3 -30 2-2

-1

a b c d2-2 3-6 4-7 -5-2

2= -1

STANDARD 10

a b c32 4

2 a b c d 56 72

-2



10

Standards 3 and 4

It is possible to add or subtract terms of a polynomial only if they are LIKE TERMS:

Simplify 5xy + 6z x -9xy + 10z x – 15z5 5 3

5xy + 6z x -9xy + 10z x – 15z5 5 3

Simplify -8a b c + 7b c - 3b c + a b c3 5 3 53 52 5

-8a b c + 7b c - 3b c + a b c3 5 3 53 52 5 = -8a b c + a b c + 7b c – 3b c3 5 3 5 3 52 5

= -7a b c + 7b c -3b c3 5 3 52 5

It is possible to use the distributive property of multiplication over addition to multiply polynomials:

Simplify 4x(2x y + 3x y – 6x y )5 4 3 32

= (4x)(2x y) + (4x)(3x y ) + (4x)(-6x y )5 4 2 3 34x(2x y + 3x y – 6x y )5 4 3 32

= (4)(2)x y + (4)(3)x y + (4)(-6)x y1+5 1+4 1+32 3

=8x y + 12x y -24x y6 5 2 4 3

= -7a b c - 3b c +7b c3 5 2 53 5

= 5xy – 9xy + 6z x + 10z x -15z5 5 3

= -4xy +16z x – 15z5 3



11

(2x +1)(x + 4)

(4) x (1)2x x+2x

+1

(4)+F O I L

= 2x + 9x + 42

= 2x + 8x + x + 42

=

Standards 3 and 4

Simplify the following expressions:

(6x +3)(2x + 5)

(5) (2x) (3)6x (2x)+6x

+3

(5)+

F O I L

=12x + 36x + 152

=12x + 30x + 6x + 152

=

(6x - 3)(x + 5)

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

+(-3)

(5)+F O I L

= 6x + 27x - 152

= 6x + 30x -3x - 152

=

(4x - 3)(3x - 7)

(-7) (3x) (-3)4x (3x) +4x

+(-3)

(-7)+

F O I L

=12x - 37x + 212

=12x -28x - 9x + 212=

First Outer Inner Last: FOIL Method.



12

Area of a Rectangle

L

A = L W

where:

W= width

L= length

A= area

W



13

(2x +1)(x + 4)

(4) x (1)2x x+2x

+1

(4)+F O I L

= 2x + 9x + 42

= 2x + 8x + x + 42

=


STANDARDMULTIPLYING POLYNOMIALS

x

x

x

1

1 1 1 1

x + 4

2x + 1



14

Simplify the following expressions:

(2x - 2)(3x - 1)

(-1) (3x) (-2)2x (3x)+2x

+(-2)

(-1)+

F O I L

= 6x - 8x + 22

= 6x -2x - 6x + 22

=



x

x

x

-1

-1

3x – 1

2x – 2

-1

x x



15

(2x – 2)(x + 3)

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

+(-2)

(3)+F O I L

= 2x + 4x – 6 2

= 2x + 6x -2x – 6 2

=



x

x

x

-1

1 1 1

x + 3

2x – 2

-1



16

(2x – 2)(x + 3)

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

+(-2)

(3)+F O I L

= 2x + 4x – 6 2

= 2x + 6x -2x – 6 2

=



x

x

x

-1

1 1 1

x + 3

2x – 2

-1



17

(2x – 2)(x + 3)

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

+(-2)

(3)+F O I L

= 2x + 4x – 6 2

= 2x + 6x -2x – 6 2

=



x

x

x

-1

1 1 1

x + 3

2x – 2

-1



18

Standards 3 and 4

x+1X

+5-3x+5x-3x2

x2

x3

+5+2xx3 -2x2

x- 4x + 7

2

x-4X

-28+16x+7x-4x2

-4x2

x3

-28+ 23xx3-8x2

x- 3x + 5

2

x- 3x + 5

2

Simplify x+1

Simplify x- 4x + 7

2

x-4



19

Standards 8, 10, 11

L

L

LB

V = Bh

B = (L)(L)

B= L2

V = L L2

V= L3

VOLUME OF A CUBE: REVIEW



20

STANDARDS

11

1

2

2

23

3

3

4

4

4

1x1x1 = 13

= 1

1 CUBED

2x2x2 = 23

= 8

2 CUBED3x3x3 = 3

3= 27

3 CUBED

4x4x4 = 43

=64

4 CUBED

What is the volume for these cubes?



21

xx

x

x

x 11

1

1 11

x

V =(x)(x)(x) V =(x)(x)(1)

V =(x)(1)(1)

V =(1)(1)(1)

= x 3 = x 2

= x

= 1

Lets find the volume for this prisms:

Can we use this knowledge to multiply polynomials?PRESENTATION CREATED BY SIMON PEREZ. All rights reserved


22

STANDARD

Multiply:

(x+3)(x+2)(x+1)

(x+2)

(x+1)

(x+3)

(x + 2)(x + 3)

(3) x (2) xx+ x

+ (2)

(3)+F O I L

= x + 5x + 6 2

= x + 3x +2x + 6 2

=

x+1X

+6+5x+6x+5x 2

x2

x3

+6+11xx3 +6x2

x +5x + 6

2



23

STANDARD

Multiply:

(x+3)(x+2)(x+1) = x + 6x + 11x + 63 2

(x + 2)(x + 3)

(3) x (2) xx+ x

+ (2)

(3)+F O I L

= x + 5x + 6 2

= x + 3x +2x + 6 2

=

x+1X

+6+5x+6x+5x 2

x2

x3

+6+11xx3 +6x2

x +5x + 6

2



24

STANDARD

Multiply:

(x+3)(x+2)(x+1) = x + 6x + 11x + 63 2

(x + 2)(x + 3)

(3) x (2) xx+ x

+ (2)

(3)+F O I L

= x + 5x + 6 2

= x + 3x +2x + 6 2

=

x+1X

+6+5x+6x+5x 2

x2

x3

+6+11xx3 +6x2

x +5x + 6

2

(x+2)

(x+1)

(x+3)

So, a third degree polynomial may be represented GEOMETRICALLY, by the VOLUME OF A RECTANGULAR PRISM, in this case with SIDES (x+3), (x+2) and (x+1).



25

STANDARD

Multiply:

(2x+1)(x+3)(x+4)

(x+3)

(x+4)

(2x+1)

(2x + 1)(x + 3)

(3) x (1) 2x x+2x

+ (1)

(3)+F O I L

= 2x + 7x + 3 2

= 2x + 6x +1x + 3 2

=

x+4X

+12+28x+ 3x+7x 2

8x2

2x 3

+12+31x2x 3+15x 2

2x +7x + 3

2



26

STANDARD

Multiply:

(2x+1)(x+3)(x+4) (2x + 1)(x + 3)

(3) x (1) 2x x+2x

+ (1)

(3)+F O I L

= 2x + 7x + 3 2

= 2x + 6x +1x + 3 2

=

x+4X

+12+28x+ 3x+7x 2

8x2

x3

2x +7x + 3

2

+12+31x2x 3+15x 2

+12+31x2x 3+15x 2=



27

STANDARD

Multiply:

(2x+1)(x+3)(x+4) (2x + 1)(x + 3)

(3) x (1) 2x x+2x

+ (1)

(3)+F O I L

= 2x + 7x + 3 2

= 2x + 6x +1x + 3 2

=

x+4X

+12+28x+ 3x+7x 2

8x2

x3

2x +7x + 3

2

+12+31x2x 3+15x 2

+12+31x2x 3+15x 2=

(x+3)

(x+4)

(2x+1)



28

Standards 3 and 4

x - 4x + 5

2

x - x -7x +153 2

x

x3 - 4x2 + 5x-

3x2 -12x +15

+3

3x2 -12x +15-

x -10x + 26

2

x -12x +46x -523 2

x

x3 -10x2 +26x-

-2x2 +20x-52

-2

2x2 +20x-52-

Divide by x - x -7x +153 2 x - 4x + 5

2

Divide by x -10x + 26

2

x -12x +46x -523 2



29

Standards 3 and 4

2 1 2 -20 24

1

2

4 -12

8 -24

0

Divide x + 2x -20x + 24 by x-2 using synthetic division23

with x- (+2)

x + 4x - 121 2

x + 4x - 122

-4 1 1 -8 16

1

-4

-3 +4

12 -16

0

Divide x +x - 8x + 16 by x+4 using synthetic division23

with x- (-4)

x - 3x + 41 2

x - 3x + 42PRESENTATION CREATED BY SIMON PEREZ. All rights reserved


30

Standards 3 and 4

Factoring the Greatest Common Factor (GCF):

4x y z - 16x y z + 32x3 23 2 4 = 4xx y z - 4(4x)xy z + 8(4x)2 3 2 4

=4x(x y z – 4xy z + 8)2 3 2 4

-27p q r + 9p q r - 3pqr3 3 2 2 2 3 = (3)(-9)pp qq r + (3)(3)ppqqrr -3pqrr2 2 2

=3pqr(-9 p q + 3pqr – r )2 2 2

= (3pqr)(-9)p q + (3pqr)(3qpr)-(3pqr)r2 2 2



31

Standards 3 and 4Difference of Two Squares:

(x+2)(x-2) x - 4=2

a - b = (a+b)(a-b)2 2

9y - 64=2 (3y+8)(3y-8)

Sum of Two Cubes:

a + b = (a+b)(a -ab + b )3 3 2 2

8y + 27z =3 3

64k +125j =3 3

Difference of Two Cubes:

a - b = (a-b)(a +ab + b )3 3 2 2

216y - z =3 3

27k - j =3 3

(2y + 3z)((2y) - (2y)(3z) + (3z) )2 2

(2y + 3z)(4y - 6yz + 9z )2 2=

(4k + 5j)((4k) - (4k)(5j) + (5j) )2 2

(4k + 5j)(16k - 20kj + 25j )2 2=

(6y - z)((6y) + (6y)(z) + (z) )2 2

(3k - j)(9k + 3kj + j )2 2=

(3k - j)((3k) + (3k)(j) + (j) )2 2

(6y - z)(36y + 6yz + z )2 2=



32

Difference of Two Squares:

(x+2)(x-2)

STANDARDSPECIAL PRODUCTS

x

x

-1

1 1

x +2

x – 2

-1



33


(x+2)(x-2)


x

x

-1

1 1

x +2

x – 2

-1



34


(x+2)(x-2) = x - 4 2


x

x

-1

1 1

x +2

x – 2

-1



35

STANDARD

x

x 1 1 1

x + 3

x – 3 -1

-1

-1

(x+3)(x-3)

SPECIAL PRODUCTS



36

STANDARD

x

x 1 1 1

x + 3

x – 3 -1

-1

-1

(x+3)(x-3)

SPECIAL PRODUCTS



37

STANDARD

x

x 1 1 1

x + 3

x – 3 -1

-1

-1

(x+3)(x-3)

SPECIAL PRODUCTS



38

STANDARD

x

x 1 1 1

x + 3

x – 3 -1

-1

-1

(x+3)(x-3) = x – 9

2

SPECIAL PRODUCTS


39

Standards 3 and 4Perfect Square Trinomials:

a + 2ab + b = (a + b)22 2

a - 2ab + b = (a - b)22 2

x + 4x + 42

= (x +2)2

x + 6x + 92

= (x +3)2

= (x) + 2(x)(2) + (2)2 2

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

25x + 40x + 162 = (5x) + 2(5x)(4) + (4) 2 2

= (5x + 4)2

x -10x + 252

= (x - 5)2

x - 14x +492

= (x -7) 2

= (x) - 2(x)(5) + (5)2 2

= (x) - 2(x)(7) + (7)2 2

64x - 64x + 162 = (8x) - 2(8x)(4) + (4) 2 2

= (8x - 4)2PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

40


(x +2)2

= (x) + 2(x)(2) + (2)2 2

x + 4x + 42=

x

x

1

1 1

x +2

x + 2

1

= (x+2)(x+2)


41


(x +3)2

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

x + 6x + 92=

x

x

1

1 1

x +3

x + 3

1

= (x+3)(x+3)

1

1


42

Standards 3 and 4General Trinomials:

B -5B -502

(B+5)(B-10)

-5-50

Two numbers that multiplied be negative fifty should be (+)(-) or (-)(+)

Two numbers that added be negative 5 should be |(-)|>|(+)|

(1)(-50) 1+(-50)= -49

(5)(-10) 5+(-10)= -5(2)(-25) 2+(-25)= -23

1

xFactor the following trinomial:


43

Standards 3 and 4

-60 -11

x

(4)(-15) 4 + -15= -11

12x - 11x -52

12x + (4-15)x -52

12x + 4x -15x -52

4x(3x)+ (4x)1 -5(3x) + (-5)(1)

4x(3x+1) – 5 (3x +1)

(4x- 5)(3x+1)

Factor the following trinomial:

Find two numbers that multiplied be (12)(-5)=-60 and added -11.

(3)(-20) 3 + -20= -17(2)(-30) 2 + -30= -28

(1)(-60) 1 + -60= -59

12x - 11x -52

General Trinomials:

acx + (ad + bc)x + bd = (ax +b)(cx +d)2


44

Standards 3 and 4

+24 +11

x

-6x +11x -42

-6x + (3+8)x -42

-6x + 3x +8x -42

-3x(2x)- (-3x)1 +4(2x) + (4)(-1)

-3x(2x-1) + 4(2x -1)

(-3x+ 4)(2x-1)


Find two numbers that multiplied be (-6)(-4)= +24 and added -11.

(3)(8) 3 + 8= 11(2)(12) 2 + 12= 14

(1)(24) 1 + 24= 25

-6x +11x - 42


45

Standards 3 and 4

-96 -4

x

(8)(-12) 8 + -12= -4

8x - 4x -122

8x + (8-12)x -122

8x + 8x -12x -122

2x(4x)+ (2x)4 -3(4x) + (-3)(4)

2x(4x+4) – 3 (4x +4)

(2x- 3)(4x+4)


Find two numbers that multiplied be (8)(-12)=-96 and added -4.

8x - 4x -122


Documents

1 Exponent Rules and Monomials Standards 3 and 4 Simplifying Monomials: Problems POLYNOMIALS Monomials and Polynomials: Adding and Multiplying Multiplying