Math 130 Test

Solving an applied problem

1. Read 2. Assign a variable 3. Write an equation 4. Solve 5. State the answer 6. Check

base

2^3 - the number 2 is the base

exponent

2^3 - the number 3 is the exponent

exponential expression (exponential)

2^3 - where the 3 indicates that three factors of 2 appear in the corresponding product

FOIL method

FOIL method. is a convenient way to find the product of two binomials .First, Outer, Inner and Last. gives the pairs of terms to be multiplied when distributing each term of the first binomial, multiplying by each term of the second binomial

Complement

Given a set A and a universal set U, the set of all elements of U that do not belong to set A is the complement of set A = A'

simple interest (I)

I = Prt - on a principal amount of P dollars at an annual interest rate r for t years

pure imaginary number

If a = 0 and b DOES NOT EQUAL 0 3i or -16i

nonreal complex number

If b DOES NOT EQUAL 0, then a + bi is a nonreal complex number 7 + 2i or -1 - i

double solution

If k = 0, then there is only one distinct solution, 0, sometimes called a double solution

Real part

In a complex number, a + bi, a is the real part

imaginary part

In a complex number, a + bi, b is the imaginary part

standard form

The form a + bi is standard form

future value (maturity value)

The formula A = P (1 + rt) which can also be written as A - P = Prt, gives the future value or maturity value, A of P dollars invested for t years at annual simple interest rate r

degree of a polynomial

The greatest degree of any term in a polynomial is the degree of the polynomial

coordinate

The number associated with a given point is the coordinate of the point

coordinate system

The number associated with a given point is the coordinate of the point. This correspondence forms a coordinate system

complex conjugate

The numbers 6 + 5i and 6 - 5i differ only in the sign of their imaginary parts and are complex conjugates. The product of a complex number and its conjugate is always a real number

Union

The union of set A and B, is made up of all the elements belonging to set A OR set B = A U B

Power rule 1

To raise a power to a power, multiply the exponents (a^m)^n = a^mxn (4^5)^3 = 4^15

set

a collection of objects

linear equation in one variable

a linear equation in one variable is an equation that can be written in the form ax + b = 0

first-degree equation

a linear equation is a first-degree equation because the greatest degree of the variable is 1

solution set

a number that is a solution of an equation is said to satisfy the equation, and the solutions of an equation make up its solution set

trinomial

a polynomial containing exactly three terms

polynomial

a polynomial is a term or a finite sum of terms, with only positive or zero integer exponents permitted on the variables

factored completely

a polynomial is factored completely when it is written as a product of prime polynomials

prime polynomial

a polynomial with variable terms that cannot be written as a product of two polynomials of lesser degree is a prime polynomial.

lowest terms

a rational expression is written in lowest terms when the GCF of its numerator and its denominator is 1

monomial

a single-term polynomial

binomial

a two- term polynomial

literal equation

an equation involving letters - example: formula

identity

an equation satisfied by every number that is a meaningful replacement for the variable is an identity 3(x + 1) = 3x + 3

contradiction

an equation that has no solution x = x + 1

conditional equation

an equation that is satisfied by some numbers but not others is a conditional equation 2x = 4

second-degree equation

an equation with a squared variable term and no terms of greater degree x^2 = 25 or 4x^2 + 4x -5

algebraic expression

any collection of numbers or variables joined by the basic operations of addition, subtraction, multiplication, or division (except by 0), or the operations of raising to powers or taking roots, formed according to the rules of algebra, is an algebraic expression -2x^2+3x, 15y/2y-3, √m^3 -64, (3a + b)^4

standard form

ax^2 + bx + c = 0

quadratic equation

ax^2 + bx + c = 0, where a, b, and c are real numbers with a cannot equal 0

complex number

complex numbers are formed by adding real numbers and multiples of i a + bi

universal set

contains all elements included in the discussion (U)

Motion problems

distance, rate ( speed or velocity) and time d = rt or r = d/t or t = d/r

equivalent equations

equations with the same solution set are equivalent equations

Zero exponent

for any nonzero real number a, a^0 = 1

Imaginary unit i

i = √(-1) and i^2 = -1

zero factory property

if a and b are complex numbers with ab = 0, then a = 0 and b = 0 or both equal zero

polynomial in x

if the terms of a polynomial contains only the variable x, then the polynomial is a polynomial in x. the terms of a polynomial cannot have variables in a denominator

square root property

if x^2 = k, then x = √k or x = -√k

number line

integers can be graphed on a number line

linear Model

is a linear equation

equation

is a statement that two expressions are equal

mathematical model

is an equation or inequality that describes the relationship between two quantities

domain of a rational expression

is the set of real numbers for which the expression is defined. because the denominator of a fraction cannot be 0, the domain consists of all real numbers except those that make the denominator 0

like terms

like terms are terms with the same variables each raised to the same powers -13x^3, 4x^3, -x^3

radical notation

n rad (a) = n is the index, a is radicand

principal nth root

n rad (a) represents a positive root, the principal nth root

irrational numbers

not rational numbers. includes √2, π

finite set

one that has a limited number of elements

descending order

polynomials in one variable are often written with their terms in descending order. the term of greatest degree is first, the one of next greatest degree is next, and so on

like radicals

radicals with the same radicand and the same index

unlike radicals

radicands are different, indexes are different

Mixture problems

rate (percent) of concentration x quantity = amount of pure substance present

infinite set

set consisting of all fractions between 0 and 1 - one that has an unending list of distinct elements

Null set or empty set

set containing no elements

set braces { }

sets are commonly written using set braces

Venn Diagram

shows a set A that is a subset of set B. The rectangle in the drawing represents the universal set of U, which is a Venn Diagram

degree of a term

the degree of a term with one variable is the exponent on the variable. for example the degree of 2x^3 is 3 and the degree of 17x is 1

conjugates

the expression a - b and a + b are conjugates

Power rule

the index of the root of a root is the product of their indexes

Intersection

the intersection of sets A and B, is made up of all the elements belonging to both set A and set B = A n B

numerical coefficient (coefficient)

the number is the numerical coefficient or just the coefficient

elements (members)

the objects that belong to a set

factoring

the process of finding polynomials whose product equals a given polynomial is called factoring

term

the product of a number and one or more variables raised to powers is a term

Multiplication property of zero

the product of a real number and 0 is 0 0 x a = a x 0 = 0

Distributive properties

the product of a real number and the sum (or difference) of two real numbers equals the sum (or difference) of the products of the first number and each of the other numbers. a (b + c) = ab + ac a (b - c) = ab - ac

Product rule

the product of two roots is the roots of the product

rational expression

the quotient of two polynomials P and Q, with Q cannot equal 0, is a rational expression

complex fraction

the quotient of two rational expressions is a complex fraction.

quotient rule

the root of a quotient is the quotient of the roots

real numbers

the set of all numbers that correspond to points on a number line. real numbers can be represented by decimals. real numbers include rational numbers

Identity Properties

the sum of a real number and 0 is a real number, and the product of a real number and 1 is that real number. There exists a unique real number 0 such that: a + 0 = a and 0 + a = a There exists a unique real number 1 such that: a x 1 = a and 1 x a = a

Inverse Properties

the sum of any real number and its negative is 0, and the product of any nonzero real number and its reciprocal is 1 There exists a unique real number -a such that: a + (-a) = 0 and -a + a = 0 if a is not 0, there exists a unique real number 1/a such that: a x 1/a = 1 and 1/a x a = 1

Associative Properties

the sum or product of three real numbers is the same no matter which two are added or multiplied first ( a + b) + c = a + ( b + c) or (ab)c = a(bc)

Closure Property

the sum or product of two real numbers is a real number (a + b is a real number. ab is a real number)

Commutative properties

the sum or product of two real numbers is the same regardless of their order (a+b = b+a, ab = ba)

absolute value

the undirected distance on a number line from a number to 0 is the absolute value of that number |a| |a| = a if a greater than or equal to 0, -a if a is less than 0

greatest common factor (GCF)

to factor a polynomial, we look for a monomial that is the GCF of three terms

Power Rule 2

to raise a product to a power, raise each factor to that power (ab)^m = a^m b^m (7x)^3 = 7^3 x^3

Power Rule 3

to raise a quotient to a power, raise the numerator and the denominator to that power (a/b)^m = a^m/b^m (3/5)^4 = 3^4/5^4

solution (root)

to solve an equation means to find all numbers that make the equation a true statement. These numbers are the solutions or roots of the equation

Disjoint sets

two sets that have no elements in common are disjoint sets

fundamental principle of fractions

we use this to write a rational expression in lowest terms by dividing out common factors

factoring by grouping

when a polynomial has more than three terms, its can sometimes be factored using factoring by grouping

Product rule

when multiplying powers of like bases keep the base and add the exponents a^m x a^n = a^m+n 2^2 x 2^3 = 2^5

Integers

{...., -3,-2,-1,0,1,2,3,...}

Whole numbers

{0,1,2,3,4,...}

Natural numbers

{1,2,3,4,....}

rational numbers

{p/q - p and q are integers and q cannot equal 0}