advanced math chapter p1 prerequisites advanced math chapter p

Advanced Math Chapter P 1

Prerequisites

Advanced Math

Chapter P


Review of Real Numbers and Their Properties

Advanced Math

Section P.1


Natural numbers

• {1, 2, 3, 4, …}


Whole numbers

• {0, 1, 2, 3, 4, …}


Integers

• { … , -3, -2, -1, 0, 1, 2, 3, … }


Rational numbers

• Can be written as the ratio p/q where q ≠ 0• Includes natural, whole, integers, and

fractions.• The decimal representation of a rational

number either terminates (like 0.25) or is repeating.


Irrational numbers

• Are not rational• Have infinite non-repeating decimal

representations.

2


You try• Which of the numbers above are…• Natural numbers?

• Whole numbers?• Integers?• Rational numbers?• Irrational numbers?

1 6 1, , , 2, 7.5, 1.8, 22

3 3 2

1, 22

6, 22

3

6

3

1 6, , 7.5, 1.8, 22

3 3

6

3


Real numbers

• Used in everyday life to describe quantities• Includes rational and irrational numbers• Doesn’t include imaginary numbers


Real number line

• Numbers to the right of origin are positive, numbers to the left are negative

• Nonnegative numbers are positive or zero• Nonpositive numbers are negative or zero

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Origin


One-to-one correspondence

• Between real numbers and points on the real number line

• Every real number corresponds to one point on the number line

• Every point on the number line corresponds to one real number


Definition of order

• If a and b are real numbers, a is less than b if b – a is positive and on a number line, a is left of b


Bounded intervals

• Have endpoints• Have finite length• See chart on page 3


Closed intervals

• Include endpoints• Shown with square brackets• “or equal to”• Open intervals don’t include endpoints

(shown with parentheses)


Open intervals

• Don’t include endpoints• Shown with parentheses


Example

• Graph the following on a number line

3 5x


You try

• Graph the following on a number line

1 3x


Unbounded intervals

• Do not have a finite length• See chart on page 4


Example

• Express the following using inequality notation

• All x in the interval (–2,4]

2 4x


You try

• Express the following using inequality notation

• t is at least 10 and less than 22

10 22t


Absolute value

• Magnitude• Distance between the origin and the point

on the number line

if 0

if 0

a a a

a a a


Properties of Absolute values

• Chart on page 5


Distance between a and b

,d a b b a a b


Variables

• Letters used to represent numbers


Algebraic expressions

• Combinations of letters and numbers

13y 3 2z

24x


Terms

• Parts of an algebraic expression separated by addition (or subtraction)


Constant term

• Term that doesn’t contain a variable


Evaluating algebraic expressions

• Substitute numerical values for each of the variables in the expression


You try

• Evaluate the following for x = 12 5 4x x


Substitution Principle

• If a = b, then a can be replaced by b in any expression involving a.


Charts

• Pages 6, 7, and 8


You try

• Exercises 98 – 104 even


Factors

• If a, b, and c are integers such that ab = c, then a and b are factors, or divisors, of c.


Prime number

• Integer that has exactly two factors: 1 and itself


Composite

• Can be written as the product of two or more prime numbers


Fundamental Theorem of Artihmetic• Every positive integer greater than 1 can be

written as the product of prime numbers in precisely one way

• Prime factorization


Exponents and Radicals

Advanced Math

Section P.2


Exponential notation

• a to the nth power• n is the exponent• a is the base

na a a a a


Properties of exponents

• Chart page 12• Read first two paragraphs on page 13


Examples

• No calculator

5

2

5

5

2

4

3

3


You try

• No calculator

033 23


Example

• Rewrite with positive exponents and simplify

222x


You try

• Rewrite with positive exponents and simplify

2 44 8y y


Scientific notation

• n is an integer• Positive exponents mean large numbers• Negative exponents mean small numbers

10nc

1 10c


Examples

• Write in scientific notation• 9,460,000,000,000• 0.00003937


You try

• Write in scientific notation• 0.0000899• 34,000,000


You try

• Write in decimal notation• 1.6022 × 10-19


Definition of nth root

• Page 15


Principal nth root

• Page 15


Tables

• Page 16


Examples

• No calculators

503

4


You try

• No calculators

2

10 12 3


A radical is simplified when

• All possible factors have been removed from the radical

• All fractions have radical-free denominators• The index of the radical is reduced


Examples

8 3 54

• No calculators


You try

• No calculators

316

2775

4


Combining radicals

• Can add or subtract if they are like radicals• Have the same index and same radicand

• Should simplify first


Example

5 9x x


You try

• No calculators

4 27 75


Rationalizing denominators

• Gets rid of radical in denominator• Multiply both numerator and denominator

by the conjugate of the denominator


Conjugates

and a b m a b m


Examples

5

10

3

5 6


You try

3

7

6

2 3


Rationalizing numerators

• Sometimes useful• Not simplifying radical• Multiply numerator and denominator by

conjugate of numerator


Rational exponents

• Definition page 19


You try

• Change from radical to rational exponent form

3 64

9


You try

• Change from rational exponent form to radical form

12144

5416


You Try

• Simplify:

512 2

3 2

5 5

5

x

x

4 2x 24 3


Polynomials and Special Products

Advanced Math

Section P.3


Polynomial

• an is the leading coefficient

• n is the degree of the polynomial• A0 is the constant term

11 1 0

n nn na x a x a x a


Example

• Coefficients are 3, 7, 8, and -5• Leading coefficient is 3• Polynomial degree 4

4 23 7 8 5x x x


Polynomials in two variables

• Degree of each term is sum of exponents• Degree of polynomial is highest degree of

its terms• leading coefficient goes with highest-degree

term

25 6 4 9xy y xy


Standard form

• Written with descending powers of x, then descending powers of y


Adding and subtracting polynomials• Add or subtract like terms (have the same

variables to the same powers) by adding and subtracting their coefficients


You try

2 3 215 6 8 14.7 17x x x


FOIL

• ONLY FOR MULTIPLYING TWO BINOMIALS

• Product of first terms + Product of outside terms + Product of inside terms + Product of last terms


You try 3 2 2 8x x


Multiplying other polynomials

• Use the distributive property• Add the products of each term of the first

polynomial times the second polynomial


Example

2 23 7 2 3x x x


Special Products

• Page 26


Factoring Polynomials

Advanced Math

Section P.4


Factoring

• Writing a polynomial as a product• Unless noted otherwise, you want factors

with integer coefficients• Completely factored when each of its

factors is prime


Removing a common factor

• Distributive property in reverse• First step in factoring a polynomial


Example32 6x x


You try

23 4 3x x


Factoring special polynomial forms• Page 34• Come from special product forms in section

P.3


Examples23 27x

28 8 2t t

3 8x


You try2 64x

23 24 48t t

38 1x


Trinomials with binomial factors

• FOIL in reverse• May involve trial and error


Examples

2 2x x

23 5 2x x


You try

2 5 6x x

29 3 2x x


Factoring by Grouping

• Sometimes works for polynomials with more than three terms• Sometimes several different options will work

• Can use to factor trinomials


Examples3 22 6 3x x x

26 2x x


You try

3 26 2 3 1x x x

22 9 9x x


Rational Expressions

Advanced Math

Section P.5


Domain

• The set of real numbers for which an algebraic expression is defined

• Usually all real numbers, except any • that make the expression equal an

imaginary number • Or make it undefined


Examples

• Find the domain of the following:

5

2

x

x

3 3x


You try

• Find the domains of the following:

3

2 1

x

x

2 2x


Simplifying Rational expressions

• Factor each polynomial completely• Divide out common factors• List the domain by the simplified

expression• The domain of the simplified expression cannot

include numbers that weren’t in the domain of the original expression


Examples

• Write the following in simplest form215

10

x

x2 16

4

y

y

3 2

3

2 3

1

y y y

y


You try

• Write the following in simplest form36

2

x

x2 5 6

2

y y

y

3 2

2

2 2

1

x x x

x


Operations with rational expressions• Factor• Then multiply, divide, add, or subtract using

the rules for fractions


Examples

5 1

1 25 2

x

x x

2 2

2

6 4

6 9 3

t t t

t t t

2 3

1 2 1

1x x x x


You try2 2

3 2 2 2

2

3 2

x xy y x

x x y x xy y

3 5

2 2x x

2

2 2 1

1 1 1x x x


Complex fractions

• Have separate fractions in the numerator, denominator, or both.

• Two ways to solve• Making one fraction in numerator and one in

denominator and dividing• Multiply numerator and denominator by LCD

of all fractions involved


Example3

4 23

2

x

x


Example

2

2

1

1

xx

x

x


You try

12

2

x

x


Simplifying expressions with negative exponents• Factor out the common factor with the

smaller exponent• When factoring, subtract exponents


Example

5 42 2 21 1x x x


You try

3 422 5 4 5x x x x


Difference Quotient

• Have a difference on the top and a constant or degree 1 term on the bottom

• In calculus, sometimes you have to rewrite them by rationalizing the numerator so that the expression is defined.


Example

1 1x h x

h


Errors and the Algebra of Calculus

Advanced Math

Section P.6


Common algebraic errors

• Read lists on your own during homework time

• Ask if you don’t understand


Algebra of Calculus

• Sometimes writing things in an “unsimplified” way makes doing calculus operations easier

• Read on your own• Let me know if you have questions

Example

• Simplify the expression


4 35 2 2 4

25

3 1 2 1 5x x x x x

x

Example

• Write the fraction as the sum of three terms


2 4 8

2

x x

x

You try

• Write the fraction as the sum of three terms


22 1x x

x


The Cartesian Plane

Advanced Math

Section P.7


Cartesian Plane

• Rectangular coordinate system• Named after René Descartes• Ordered pair: (x, y)• Horizontal x-axis• Vertical y-axis• Origin: where axes intersect


QuadrantsMath Composer 1. 1. 5http: / /www. mathcomposer. com

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-5

-4

-3

-2

-1

1

2

3

4

5

x

y

I

IVIII

II


Scatter plots

• Each point is plotted• Dots are not connected


Distance formula

2 222 1 2 1d x x y y

2 2

2 1 2 1d x x y y

2 2

2 1 2 1d x x y y

Math Composer 1. 1. 5http: / /www. mathcomposer. com

(x1, y1)

(x1, y2) (x2, y2) x

y

• Pythagorean theorem

d

2 2 2a b c


You try

• Find the distance between (-3, 2) and (3, -2)


Verifying a right triangle

• Showing that three given points are vertices of a right triangle.

• Plot the points• Use the distance formula to find the

distances between the points.• See if the distances work in the Pythagorean

theorem


Example

• Use the distance formula to show that the points (9,4), (9,1), and (-1,1) form a right triangle.


Midpoint formula

• To find the midpoint of the line segment joining two points, average the x-coordinates and average the y-coordinates.

• Midpoint has coordinates

1 2 1 2,2 2

x x y y


Example

• Find the midpoint of the segment joining the points (1, 1) and (9, 7).


You try

• Find the midpoint of the line segment joining the points (–1, 2) and (5, 4).


Example

• Use the midpoint formula to find points that divide the line segment joining (1, –2) and (4, –1) into four equal parts.

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