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Chapter 5 Algebra 2H 1
Multiplying Monomials
Monomial ( ) –
(3x2)(2x
4) = (3x
2)
3 =
(2xy4)(-3x
2y
2z) = (xy)(x
2y)
4 = [(3x
2y)
2]
5 =
x2y
(x3y+1
) = (y2n
)3 = (x
2n)
4n =
(x2z
4)(2xyz
4)(-3x
3y
2) =
Negative Exponents
Negative exponents:
3x 22x
3
1
x
1
2
x
2
1
3x
Chapter 5 Algebra 2H 2
Dividing Monomials
Rule: Unless otherwise stated all answers will have positive exponents
10
3
x
x
2
8
8x
2x
2
4
2x
y
0x 02
03y Rule: No 0 exponents in answer
4 4x x 4
23x
2
2 4x y
2 22x y xy
2 3
4
36a b
3ab
22 3
32
4x y
2xy
23 4
2
x y
x y
Homework
Page 226 18 – 39 column
Chapter 5 Algebra 2H 3
Scientific Notation
24,300,000 = 0.000000045
4.13 X 105 = 1.1 X 10
-8 =
(2.3 X 107)(1.3 X 10
5) = (3200000000)(0.0000005) =
On calculator:
Homework
Page 227 44, 47, 51, 53, 55, 70
Polynomials
Polynomial –
# of terms –
Binomial –
Trinomial –
Descending order –
Degree –
Chapter 5 Algebra 2H 4
Leading coefficient –
Constant term –
Quadratic –
Cubic –
For the following polynomials state :
# of terms
Degree of polynomial
Leading coefficient
Constant (if any)
3x2 + 2x + 7 -4x
2 + 3 – 3x
4 + 9x
Create a cubic, trinomial, with a leading coefficient of 2 and a constant term of -4.
Adding/Subtracting Polynomials
Rule: All polynomials, in one variable, will be written in descending order!
(3x2 – 2x + 7) + (-3x
2 + 2x – 12) =
Chapter 5 Algebra 2H 5
(2x2 + 3x – 7) – (5x
2 – 1 – 8x) = (3a
2 – 9a) – (-5a
2 + 7a – 6) =
Multiplying Polynomials by a Monomial
2x(2x + 4) = -4x2(4x – 6x
2) =
x – 2x(x – 2) = -2x2(3 – 2x – 3x
2 + 2x
3) =
3xy2(3x
2 – 2xy + 4y
2) = 3x
2 –x[x – 2(3x – 4)] =
Homework
Page 231 23 – 31 odd
Chapter 5 Algebra 2H 6
Multiplying Polynomials
(2x – 3)(2x + 5) = (3x + y)(2x2 – 3y) =
(xy + 4)(xy – 3) = (x – 2)(x2 – 3x + 7) =
(x2 + 2x – 3)(x
2 – 5x + 7) = (4x + 2)
2 =
3x(4x + 1)(x2 – 5x + 2) =
Homework
Page 232 41 – 50, 65
Chapter 5 Algebra 2H 7
Dividing Polynomials by Monomials
Divide: (different than simplify)
3 4 5
2
2x 4x 10x
2x
2 2 3 2
2
10x y 16x y 7x
4x y
Dividing Polynomials
Divide:
2(x 14x 24) (x 2) 2x 14x 24
x 2
3 2(6x 4x 8) (2x 4) 218x 3x 2
3x 2
Homework
Page 328 15, 17, 19, 23, 31, 39, 41, 45, 68
Chapter 5 Algebra 2H 8
Synthetic Division
If you dividing a polynomial by a linear binomial, you can use synthetic division. You want the
linear term to be ______________.
2(3x 19x 20) (x 5) 2(3x 15) (x 3)
4 2(8x 4x x 4) (2x 1)
Homework
Page 237 23, 31, 39, 45
Factoring
G.C.F.
4x3y
2 + 12x
3 + 20xy
2 32x
2 + 12x
9x2 + 14y
4 x
2y
4 – x
2y – 4x
2
Chapter 5 Algebra 2H 9
Grouping:
a3 – 4a
2 + 3a – 12 3x
3 – 5 + x
2 – 15x
Homework
Factoring handout 1 - 9
Quadratic Trinomial
x2 + 12x + 20 y
2 – 18y + 72
2x2 + 8x – 64 x
4 – 6x
2 – 16
Chapter 5 Algebra 2H 10
5x2 – 13x + 6 x
2 – 5x + 7
2x2 – 11x – 21 6 + x – 12x
2
Nobes Method
2x2 + 7x + 3 Steps:
Chapter 5 Algebra 2H 11
6x2 + 17x – 10 12 – x – 6x
2
4x2 – 42xy + 20y
2 Find all, if any, integer values for k such that
2x2 – kx – 5
Homework
Factoring handout 10 - 21
Difference of two perfect squares
Perfect squares:
A variable is a perfect square if...
x2 – 4 25x
2 – 1
Chapter 5 Algebra 2H 12
General Pattern a2 – b
2
36x4 – y
6 16 – x
10
x4 – 16 3x
2y
4 – 12
25x2 + 16 49y
2 – 81
Sum or Difference of two perfect cubes
Perfect cubes:
A variable is a perfect cube if...
x3 – 8 27 – y
6
Chapter 5 Algebra 2H 13
General Patterns:
a3 – b
3 a
3 + b
3
64x3 + 1 216 – 125b
9
8x3y
6 + 27 16x
3 – 9
Homework
Factoring handout 22 - 33
Chapter 5 Algebra 2H 14
x3 – 2x
2 – x + 2 x
4y
2 - 5x
3y
3 + 6x
2y
4
x4 – 5x
2 – 4 a
3b
6 – b
3
Homework
Page 242 15 – 41 odd
Roots and Radicals
Simplify:
7 9 9 9
Chapter 5 Algebra 2H 15
9 3 8 3 8 3 8
3 8 2x 3x 4x
5x 3 2x 3 3x 3 4x
3 5x 3 6x 48 180
9 624a b 7 1260xy z
4 532 24x y
Homework
Page 248 36,37,44,45
Page 254 15 - 26
Chapter 5 Algebra 2H 16
Add/Subtract Radical Expressions
Add/Sub and simplify
5 5 3 7 7 3 y 12 y
6 5 35 5 x 6 2 6
27 75 35x 8x 2 50x
Homework
Page 254 35 - 38
Multiplication/Division of Radical Expressions
Mult/Divide and simplify
3 3 2 6 3 4a b ab 3 3 4 3
Chapter 5 Algebra 2H 17
3 3 5 2
x 2
x 2 x 5 2 3 2 4
Homework
Page 254 39 - 42
Rationalizing
12
6
3x
y
- No fractions under a
- No radaicals in the of a fraction.
Chapter 5 Algebra 2H 18
Simplify (rationalize)
3
6
x
3
2
6y
3
5
9
Homework
Page 254 25 - 30
Simplify
5
2 7
7
x 3
3 x
3 x
Homework
Page 254 43 – 48
Chapter 5 Algebra 2H 19
Rational Exponents
a
bx b ax
x 3 x 3x 5 2x
Simplify:
2
327 24
2
532
3
5243
1 5
3 3x x
1
3
4
3
x
x
= 8
6
81
3
24 9z
33 5 5 3 36
Homework
Page 261 23, 28, 34, 41-63 odd
Chapter 5 Algebra 2H 20
Domains of Radical Expressions
Find the domain:
3y
x 2
y 3x y x 2
y 3 4 2x y 2x x
Solving Radical Equations and Inequalities
Solve:
x 18 0 x 2 6 0 3 x 3
3x 4 3 x 2 2x 2 3 1
Chapter 5 Algebra 2H 21
1 4x 3 5 3 x 2 3
4a 1 3 4a 2 2x 3 4 5
y 9 y 3
Homework
Page 266 19, 21, 23, 27, 31, 33, 50, 51
Chapter 5 Algebra 2H 22
Imaginary Numbers
4 4 1 i2 =
4 12 2 550x y
3 64
i3 = i
23 = 4i(-2i)
2 = (
Complex Numbers
A complex number is a real number ± an imaginary number
4 + 3i 2 – i 6 3i
Note: all numbers, real and imaginary, are part of the complex number system.
Simplify and write as a complex number:
4 8i
3
3i – 4
Chapter 5 Algebra 2H 23
5 9 60 48
225b 48b
Homework
Page 274 18 - 29
Adding and Subtracting Complex Numbers
Remember to write all answers in proper complex number form a ± bi
(2 + 4i) + ( 6 – 5i) = 8 4 2 16
(-2 – 4i) – ( 6 – 8i) = (5 – 3i) + 2i =
Chapter 5 Algebra 2H 24
(7 + 2i) + ( -7 – 2i) =
Homework
Page 274 30 - 33
Multiplying Complex Numbers
(7i)(-9i) = 2 8
2i( 6 + 2i) = 2 8 2
(5 – 2i)(3 + i) = 6 3 2 1
i i5 5 3 3
Chapter 5 Algebra 2H 25
(3 – i)( 3 + i) = (5 – 2i)2 =
Homework
Page 274 34 – 37, 42 , 43
Divide/Simplify Complex Numbers
Sort of like “rationalizing”: we do not want “i” in the denominator
3
i
2 3i
4i
4
5 i
2 3i
4 i
Homework
Page 274 38 – 41, 44 - 47
Chapter 5 Algebra 2H 26
Solve for x:
2x2 + 12 = 0 -2x
2 = 80
Find m and n
(2m + 5) + (1 – n)I = -2 + 4i (m + 2n) + (2m – n)I = 5 + 5i
Homework
Page 274 49 – 61 odd
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