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}
