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
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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:
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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.
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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
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5
Standards 3 and 4MONOMIALS
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
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6
Standards 3 and 4MONOMIALS
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
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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:
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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:
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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
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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
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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.
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12
Area of a Rectangle
L
A = L W
where:
W= width
L= length
A= area
W
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13
(2x +1)(x + 4)
(4) x (1)2x x+2x
+1
(4)+F O I L
= 2x + 9x + 42
= 2x + 8x + x + 42
=
First Outer Inner Last: FOIL Method.
STANDARDMULTIPLYING POLYNOMIALS
x
x
x
1
1 1 1 1
x + 4
2x + 1
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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
=
First Outer Inner Last: FOIL Method.
STANDARDMULTIPLYING POLYNOMIALS
x
x
x
-1
-1
3x – 1
2x – 2
-1
x x
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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
=
First Outer Inner Last: FOIL Method.
STANDARDMULTIPLYING POLYNOMIALS
x
x
x
-1
1 1 1
x + 3
2x – 2
-1
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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
=
First Outer Inner Last: FOIL Method.
STANDARDMULTIPLYING POLYNOMIALS
x
x
x
-1
1 1 1
x + 3
2x – 2
-1
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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
=
First Outer Inner Last: FOIL Method.
STANDARDMULTIPLYING POLYNOMIALS
x
x
x
-1
1 1 1
x + 3
2x – 2
-1
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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
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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
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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?
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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
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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
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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).
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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
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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=
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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)
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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
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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
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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=
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32
Difference of Two Squares:
(x+2)(x-2)
STANDARDSPECIAL PRODUCTS
x
x
-1
1 1
x +2
x – 2
-1
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33
Difference of Two Squares:
(x+2)(x-2)
STANDARDSPECIAL PRODUCTS
x
x
-1
1 1
x +2
x – 2
-1
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34
Difference of Two Squares:
(x+2)(x-2) = x - 4 2
STANDARDSPECIAL PRODUCTS
x
x
-1
1 1
x +2
x – 2
-1
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35
STANDARD
x
x 1 1 1
x + 3
x – 3 -1
-1
-1
(x+3)(x-3)
SPECIAL PRODUCTS
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36
STANDARD
x
x 1 1 1
x + 3
x – 3 -1
-1
-1
(x+3)(x-3)
SPECIAL PRODUCTS
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37
STANDARD
x
x 1 1 1
x + 3
x – 3 -1
-1
-1
(x+3)(x-3)
SPECIAL PRODUCTS
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38
STANDARD
x
x 1 1 1
x + 3
x – 3 -1
-1
-1
(x+3)(x-3) = x – 9
2
SPECIAL PRODUCTS
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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
STANDARDSPECIAL PRODUCTS
(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)
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41
STANDARDSPECIAL PRODUCTS
(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
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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:
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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
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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)
Factor the following trinomial:
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
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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)
Factor the following trinomial:
Find two numbers that multiplied be (8)(-12)=-96 and added -4.
8x - 4x -122
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