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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


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

26 18 – 39 column


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 –


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) =


(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


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


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


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


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


5x2 – 13x + 6 x

2 – 5x + 7

2x2 – 11x – 21 6 + x – 12x

2

Nobes Method

2x2 + 7x + 3 Steps:


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


Difference of two perfect squares

Perfect squares:

A variable is a perfect square if...

x2 – 4 25x

2 – 1


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


General Patterns:

a3 – b

3 a

3 + b

3

64x3 + 1 216 – 125b

9

8x3y

6 + 27 16x

3 – 9

Homework



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


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


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


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.


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


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


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


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


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


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 =


(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


(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


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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