Math 1311 Business Math I

Math 1311 – Business Math I

Day 1: Review

Sets of Numbers

____ = { 1, 2, 3, 4, ..... } is called the set of ______.

Let x be one of these numbers. Find the solution of the following equations.

2x + 4 = 6 è x = ______x + 2 = 2 è x = ______

_____ = { 0, 1, 2, 3, ... } is called the set of ______

Let x be one of these numbers ( ) . Find the solution of the following equations.

x – 4 = 6 è x = ______x + 6 = 2 , è x = ______

______= { ..., -3, -2, -1, 0, 1, 2, 3, ... } is called the set of ______

Let x be one of these numbers. Find the solution of the following equations.

x – 4 = - 6 è x = _____ 2x + 1 = 2 è x = ______

Other Sets and other names for the sets above

With respect to integers the set { 1, 2, 3, ... } is also called the set of ______

With respect to integers the set { 0, 1, 2, 3, 4, .... } is also called the ______

{ 2, 3, 5, 7, 11, ... } represents the set of ______

If a natural number is not prime and is greater than 1, then it is called ______

The set that contains all real numbers that can be written as fractions is called the set of ______

We describe this set in what is called set-builder notation: Q = { a / b | a is an integer and b is a nonzero integer}

Real numbers that are not rational numbers are called ______

( this means the real numbers are made up entirely of rational and irrational numbers)

If a real number can not be written as a fraction – we call it an ______

Rational: ______Irrational:______

Real numbers

Rational Irrational #’s

pure fractions integers

negative integers nonnegative integers

(whole numbers)

zero positive integers

(natural numbers, counting numbers)

Properties of Real Numbers

commutative law of addition commutative law of multiplication

a + b = ______ab = ______

examples:

( 3 ) + ( - 4 ) = ______( 2/3 ) • ( 5/7) = ______

associative law of addition associative law of multiplication

a + ( b + c ) = ______( a b ) c = ______

examples:

1.3 + ( 3.7 + 4.8) = ______( • ) •

Distributive law of multiplication over addition

a(b + c ) = ______

examples:

3 • ( -2 + x ) = ______1.7 ( 2.9 – 1.2 ) = ______

commutative and the associative laws do not hold true with subtraction and division – do they ?

ex. 2 – 4 = 4 - 2 ? ______12 ¸ 6 = 6 ¸ 12 ? ______

ex. ( 12 – 4 ) - 2 = 12 - ( 4 – 2 ) ? ______(16 ¸ 8 ) ¸ 2 = 16 ¸ ( 8 ¸ 2 ) ? ______

Other terms

Given a natural number(greater than 1): the number is either prime or it is composite. If it is composite, it can be written as a

product of prime numbers ( prime factors). The product is called the prime factorization of the number.

ex. 24 = ______ex. 150 = ______

ex. 95 = ______ex. 29 = ______

GCF - greatest common factor

Given two or more natural numbers we can find the greatest ( largest) number that will divide evenly into all of the given

numbers. This is called the greatest common factor – GCF ( or sometimes: the GCD – greatest common divisor)

Find GCF of

12 and 20è ______( 8, 20, 30 ) = ______

248 and 324

LCM – least common multiple

Given two or more natural numbers we can find the smallest (least) number that all the given

numbers will divide evenly into.

Find LCM ( 20, 15 ) = ______LCM ( 2, 3, 5 )

Find GCF and LCM of 150 and 240. ______

Absolute Value of a real number: Let x be a real number. The absolute value of x, written | x |, is defined by

x if x > 0

| x | = either

- x if x < 0

ex. | 6 | = ______| - 2 | = ______- | - 4 | = ______,

| x | = ______if x is a natural number

Inequalities

< : means less than £ : means less than or equal to ex. 4 < 6 è we say 4 is less than 6

> : means greater than ³ : means greater than or equal to ex. 12 ³ 10 è we say 12 is greater than or

equal to 10

True or False

4 < - 4 - 2 £ - 2 3/ 11 > 2 / 9 0.01 < 0.0091

______

Using inequalities we can represent numbers on a number line –

x < 4 x ³ 2 - 4 < x £ 2

Integer Exponents

xn : exponential notation, x is called the ______and n is called the ______or the ______

and xn = x·x·x·x ·…· x ( a total of n x’s) – this is the expanded notation

ex. 25 = ______( - 4) 3 = ______- ( 4)2 = ______

ex. – 3 · 3 · 3 · 3 = ______ex. – 42 = ______( 2/3 )4 = ______

ex. 0.014 = ______

Def. If x is a real number not equal to zero, then xo = ______

( 4)0 = ______( - 2 ) 0 = ______, ( - c) 0 = ______for c ¹ 0 ( 0) 4 = ______

What about - ( 4 ) 0 = ______So, - 3 0 = ______and - ( - 2 )0 = ______. What is 0 0 = ______

Def. If n is a natural number, then x – n = ______. This gives us a way to work with negative exponents in terms of

natural numbers.

ex. Find

2-2 = ______4-3 = ______8-1 = ______

What about ( ¼) – 2 = ______( ½) – 3 = ______( ¾)-2 = ______

Properties of exponents. Let x and y represent real numbers and let m and n be integers.

1. xn · xm = ______x4 · x5 = ______x8 · x = ______

2. xn ¸ xm = ______x 8 ¸ x4 = ______x12 / x20 = ______

3. (xn ) m = ______(x3 )2 = ______(x6 )3 = ______

4. ( x · y ) n = ______(2x)3 = ______( 4xy)3 = ______

( x2y3 )2 = ______( 2xy3 )3 = ______

5. ( x ¸ y ) n = ______( x/y)4 = ______( x2 / y3 ) 3 = ______

Other examples.

1) ( 2x3 y) · ( - 4x2y4 ) = ______2) ( 2xy2 )2 · ( 3x2y )3 = ______

4xy - 2y3

3) ------¸ ------= ______

12xy6 x2y

-2x 3xy3

4) ( ------)2 ¸ ( ------)2 = ______

y2 -2y3

Since we have a definition for negative exponents – how would we work problems with negative exponents.

1. ( x - 2)- 4 = ______( x-3y-4)-2 = ______

( x-3 )2 = ______( 2x-2 )-2 = ______

2. ( 2x-3y0 ) · ( 3x-1y ) = ______

Since the rules work with integral exponents – do they work with rational exponents ? Yes

1) x 2/3 · x7/3 = ______2) x 1/5 y · x y 2/3 = ______3) ( x -2/3 ) –6 = ______

4) ( - 5x1/2y)1/2 = ______

More Examples of exponent problems –

Find

( - 2xy3 ) 0 = ______if neither x nor y = 0 180 = ______- 40 = ______

0 4 = ______0 - 2 = ______00 = ______- 42 = ______- 4 – 2 = ______

Also, if x = -1 and y = - 2 and z = 0 find

1) xy ______2) y0 = ______3) - x2 = ______4) (x – y ) ¸ x + y = ______

Radicals

square roots: of 49 ==> ______square roots of 81 ==> ______

Cube roots of 64 ==> ______cube roots of - 27 ==> ______

Continue by asking for 4th roots, 5th roots,….Notice that each number has either 1, 2, or none nth roots.

ex. Find the square roots of 25 è ______Find the fifth roots of – 32 è ______

Find the 4th roots of ( -16) è ______

n ___

Define Principal nth roots of x: \/ x ; n is called the ______, x is the ______

n __ 4 ____

if x > 0, then \/ x > 0 è \/ 125 = ______

n __ 3 ____

if x < 0 and n is odd, then \/ x < 0 è \/ - 27 = ______

n __ 2 ___

if x < 0 and n is even, then \/ x has no real value è \/ -4 = ______

ex. Find each of the following nth roots

___ 3 ___ 2 ____

1) \/ 64 = ______2) \/ - 8 = ______3 ) \/ - 25 = ______

4 ___ 2 _____ 3 _____

4) \/ x8 = ______5) \/ x6y4 = ______6) \/ 8x3y12 = ______

______

True or False: \/ x2 = x ______So, \/ ( -2)2 = ______

___

However, if we assume that x is a positive real number, then \/ x2 = x

n ___

Define x 1/n = \/ x

Examples of x1/ n

1) 8 1/3 = ______2) 16 ¼ = ______

3) - 25 ½ = ______4) ( - 27 ) 1/3 = ______

5) ( - 9 ) ½ = ______6) ( 16) – 1 / 4 = ______

n ___ n __ m

Define x m/n = \/ x m or ( \/ x )

Now we can use fractional exponents: 16 ¾ = ______, - 9 3/2 = ______, - 16-1/2 = ______

ex. (16x2/3y9 ) 1/ 4 = ______ex. ( -8x6y9 ) 2/3 = ______

ex. ( 4x-4y8 ) – ¾ = ______

Polynomials

(factors, terms, degree)

Sum of literal expressions in which each term consists of a product of constants and variables with the restriction that each

variable must have a nonnegative integer exponent.

2x, 3x2y – 4, 1 + x + 3x2, 5 – 3xy + y9, …. è What about ______or ______?

Special types of polynomials

if one term: ______if two terms: ______if three terms: ______

How many terms does each of the polynomials have ?

2xy 3 + 2xy + x9 + y2 1 + x + x2

______

Def. (degree)

monomials : 3x ==> ______2x5 ==> ______½ x5 y3 ==> ______

binomials: 2x – 1 ==> ______x + y ==> ______3x2y - 4x3y2 ==> ______

other polynomials: x – 2xy + y3 ==> ______x10 - 2xy + x6y5 ==> ______-

Basic operations of polynomials: sum – difference, products - quotients

Find ( sum and differences)

a) (2xy - 4x ) + ( 3xy - 2x ) ==> ______

b) ( 3x2 - 2x + 3 ) - ( 2x2 - 4x - 5 ) ==> ______

Products

a) 2(x – 3y ) = ______b) x( x + 2y ) = ______

c) 3x3 ( 3xy – 2x4y ) = ______

d) ( 2x –3y )( 4x + 2y ) = ______

Def. The process of writing a polynomial as a product of other polynomials of equal or lesser degree is called

______

Recall: GCF

Find GCF ( 20, 36 ) = ______GCF ( xy, x ) = ______GCF(2xy, 6y2 ) = ______

GCF ( 12x3y6, 8x4y2 ) = ______

Special Products and Factoring

1) greatest common factor: x( y + x) = ______

2) difference of squares : (x – y ) ( x + y ) = ______

3) sum – difference of cubes : ( x – y ) ( x2 + xy + y2 ) = ______

4) perfect squares: (x + y )2 = ______= ______

5) Trinomials of the form ax2 + bx + c

Review of methods of factoring -

Factoring: process of writing a polynomial as a product of other polynomials of equal or lesser degree.

Methods:

GCF – always look for a common factor - 1st method

Difference of squares: x2 - y2 = (x – y ) ( x + y)

Sum – Difference of Cubes: x3 + y3 = ( x + y ) ( x2 - xy + y2 ) , x3 - y3 = ( x – y ) ( x2 + xy + y2 ) -- SOPPS

Perfect Squares: x2 + 2xy + y2, first and last must always be positive

Multiply (x + 2y )2 = ______(3x – 4y )2 = ______

Factor: x2 - 8x + 64 = ______x2 + 10x + 25 = ______

x2 + 4xy + 4y2 = ______4x2 - 12xy + 9y2 = ______

x2 - 16x - 16 = ______

Grouping –

2 ( x – y ) - y ( x – y ) = ______xy + 2x - y2 - 2y = ______

x2 - y2 – 2y - 4 = ______x3 - y3 - x + y = ______