# College Algebra 2 Theory Definitions

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What is a Natural Number?
Any non-decimal number greater than 0, not including 0.
What is a Whole Number
All Natural numbers, and the number 0.
What is an integer?
All positive and negative natural numbers and the number 0.
What is a Rational Number?
All numbers which can be expressed in the form, where a and b are integers and b is not equal to 0 (b <> 0). Rational numbers can be expressed as terminating or repeating decimals.
What is an Irrational Number?
All numbers whose decimal representations are neither terminating nor repeating. Cannot be expressed as a quotient of integers.
What is the property of the |a|?
It is greater than or equal to 0. As in |a|>=0.
What is equal to |-a|?
|a|
Is the number a greater than, less than and/or equal to |a|.
<=
|ab| is the same as?
|a| |b|
The absolute value of a/b is the same as?
|a|/|b|
|a+b| is less than, greater than and/or equal to |a|+|b|? What is this called?
<=
This is called the Triangle Inequality.
What is the distance between two points (a & b) on the Real Number Line?
|a-b| or |b-a|
Or
The absolute value of the difference between the 2 real number.
What is the Order of Operations Agreement?
1. Innermost parentheses and work outward. If a fraction, treat the fraction as though it were in parentheses.
2. All Exponential expressions
3. Multiplication and division, left to right
4. Addition and Subtraction, left to right.
What is Commutative Property of Addition?
Two real numbers can be added in any order.
a + b = b + a
What is the Commutative Property of Multiplication?
Two real numbers can be multiplied in any order.
ab = ba
What is the Associative Property of Addition?
If three real numbers are added, it makes no difference which two are added first.
(a+b)+c = a+(b+c)
What is the Associative Property of Multiplication.
If three real numbers are multiplied, it makes no difference which two are multiplied first.
(ab)c = a(bc)
What is the Distributive Property of Multiplication over Addition?
a(b+c) = ab+ac
What is the Identity Property of Addition?
Zero can be deleted from a sum.
a + 0 = a
What is Identity Property of Multiplication?
One can be deleted from a product.
a*1=a
What is the Inverse Property of Addition?
The sum of a real number and its additive inverse gives 0, its additive identity.
a + (-a) = 0
What is the Inverse Property of Multiplication?
The product of a nonzero real number and it multiplicative inverse gives 1, the multiplicative identity.
a * 1/a = 1, a <> 0
What is a Natural Number Exponent?
If b is a real number and n is a natural number, then b^n is n factors of b. Furthermore b^1 = b
What is the Negative Exponent Rule?
If b is any real number other than 0 and n is a natural number, then b^-n is equal to 1/b^n.
What is the Zero Exponent Rule?
If b is any real number other than 0, then b^0 is equal to 1.
b^0 = 1
What is the Product Rule?
When multiplying exponential expressions with the same base, add the exponents. Use this sum as the exponent of the common base.
b^m * b^n = b^(m+n)
What is the Power Rule (Powers to Powers)?
When an exponential expression is raised to a power, multiply the exponents. Remove the parentheses and use the product for the exponent on the base.
(b^m)^n = b^mn
What is the Quotient Rule?
When dividing exponential expressions with the same nonzero base, subtract the exponent in the denominator from the exponent in the numerator. Use this difference as the exponent of the common base.
b^m/b^n = b^(m-n), b<>0
What is the Products to Powers rule?
When a product is raised to a power, raise each factor to that power.
(ab)^n = a^n * b^n
What is the Quotients to Powers rule?
When a quotient is raised to a power, raise the numerator to that power and divide by the denominator to that power.
(a/b)^n = a^n/b^n, b<>0
What is the Scientific Notation form of a number?
A number expressed as greater than or equal to 1 and less than 10, multiplied by some power of 10.
Ax10^n=A (in decimal notation)
What is the square root symbol called?
What is the Principal Square Root?
If a is a nonnegative real number, the nonnegative number b such that b^2=a, denoted b = SqRt(a), is the princpal square root of a.
What's an example of a Square Root of a Perfect Square?
SqRt(a^2) = |a|
What is the Product Rule for Square Roots?
If a and b represent nonnegative real numbers, then
SqRt(ab) = SqRt(a)SqRt(b) and SqRt(a)SqRt(b) = SqRt(ab)
What is the Quotient Rule for Square Roots?
If a and b represent nonnegative real numbers and b <> 0, then
SqRt(a)/SqRt(b) = SqRt(a/b) and SqRt(a/b) = SqRt(a)/SqRt(b)
Two or more square roots that share the same Radicand.
The number in the Radical Sign.
The number a is a radicand in:
SqRt(a)
What can be done with 2 Square Roots when you try to Add or Subtract them?
If the have Like Radicals, then you can combine them.
a*SqRt(x) + b*SqRt(x) = (a+b)*SqRt(x)
How do you Rationalize the Denominator in a Radical Expression?
You rewrite a radical expression as an equivalent expression in which the denominator no longer contains a radical.
1/SqRt(x) =
1/SqRt(x) * SqRt(x)/SqRt(x) =
SqRt(x)/SqRt(x^2) =
SqRt(x)/x
How do you Rationalize Denominators when there are more than one terms in the denominator?
You should Multiply by the opposite of the denominator such that,
Denominator is x + SqRt(y), then multiply by x - SqRt(y). Or SqRt(x) - y would be multiplied by SqRt(x) + y.
What is the Principal nth Root of a Real Number?
nthRt(a)=b means that b^n = a
If n is even then a is nonnegative (a>=0) and b is also nonnegative (b>=0). If n is odd, then a and b can be any real number.
What is the Index of an nth Root?
It is the number represented by n. Such that 'b'thRt(a), b is the index.
What is a Perfect nth Power?
A number that is the nth power of a rational number. As in 8 is a perfect third power, or perfect cube, because 8 = 2^3.
How do you find nth roots of perfect nth powers?
In general terms:
If n is odd, then nthRt(a^n) = a
If n is even, then nthRt(a^n) = |a|
True or False: the Product and Quotient Rules don't apply for cube roots, fourth roots, and all higher roots?
False
What is a Rational Exponent?
If nthRt(a) represents a real number and n>=2 is an integer, then
a^1/n = nthRt(a)
a^-1/n = 1/a^1/n = 1/nthRt(a), a<>0
How do you represent a^m/n as a Rational Exponent?
If nthRt(a) represents a real number, m/n is a rational number reduced to lowest terms, and n>=2, then
a^m/n = (nthRt(a))^m = nthRt(a^m)
OR a^-m/n = 1/(a^m/n)
How do you reduce the Index of a Radical.
Convert nthRt(x^m) to x^m/n, then reduce the fraction. Then you can return the Exponential Expression to a Radical, if necessary.
9thRt(x^3) = x^3/9 = x^1/3 = CubeRt(x)
True or False: Whole numbers are a subset of Natural Numbers.
False. Natural Numbers are a subset of Whole Numbers.
Denote the sets and subets (c) of numbers.
N (Natural) c W (Whole)
W c Z (Integer)
Z c Q (Rational)
Q U I (Irrational) ~ R (Real Numbers)
When will the decimal form of the fraction 1/n experience a repeating decimal?
When you get to the n-1 decimal place. Such that the decimal place of 1/7 will repeat at the 6th decimal place.
What kind of number do you get when you combine or add a Real number to an Imginary number i?
You get a Complex Number (C).
What is a Polynomial?
A single term or the sum of two or more terms containing variables with whole number exponents.
How do you express a polynomial in Standard Form?
When you write the terms in the order of descending powers of the variable.
What is the degree of the monomial ax^n?
n
Explain the names of simplified polynomials such as monomial, binomial and trinomial.
A monomial is a simplified polynomial with one term (ax^n). A binomial has two terms. A trinomial has three terms. A simplified polynomial with four or more terms has no special name.
What is the degree of a polynomial?
The highest degree of all terms. The degree of ax^n+bx^m+cx^p is n when n>m>p. The degree of a(x^n)(y^m) is n+m.
Group like terms, combine like terms and simplify.
How do you subtract polynomials?
Rewrite subtraction as addition of the additive inverse. Insure that you change the sign of all terms inside the parentheses preceded by the negative sign. Group like terms, combine like terms and simplify.
What is the FOIL method of finding the product of two binomials?
F represents the product of the first terms in each binomial, O is the product of the outside terms, I is the product of the two inside terms and L is the product of the last, or second, terms in each binomial.
What is the product of the sum and difference of two terms? Such as (A+B)(A-B).
The square of the first term minus the square of the second. As in A^2 - B^2.
What is the square of a binomial sum?
Such as (A + B)^2.
The first term squared plus two times the product of the terms plus the last term squared. As in A^2 + 2AB + B^2.
What is the square of a binomial difference? Such as (A - B)^2.
The first term squared minus two times the product of the terms plus the last term squared. As in A^2 - 2AB + B^2.
What is the cube of a binomial sum? Such as (A + B)^3.
The first term cubed plus three times the product of the first term squared times the second term plus three times the product of the first term times the second term squared plus the last term cubed. As in A^3 + 3A^2B + 3AB^2 + B^3.
What is the cube of a binomial difference? Such as (A - B)^3
The first term cubed minus three times the product of the first term squared times the second term plus three times the product of the first term times the second term squared minus the last term cubed. As in A^3 - 3A^2B + 3AB^2 - B^3.
True or False:
Adding, subtracting and multiplying polynomials with two or more variables is different than doing it with polynomials with one variable.
False. Its the same. In adding and subtracting you Group like terms (after adding the additive inverse when subtracting), combine like terms and simplify. In multiplication, you use FOIL.
How do you factor by grouping.
Group terms that have a common factor, factor out the greatest common factor form the grouped terms. Factor out of both terms.
What is the strategy for factoring a polynomial?
If there is a common factor, factor out the GCF. Find the number of terms in the polynomial and try factoring as follows:
If there are two terms, use on of the special forms (Difference of 2 squares, Sum/difference of two cubes).
If there are 3 terms, is it a perfect square trinomial?
Check to see if any factors with more than one term in the factored polynomial can be factored further.
How do you express numbers excluded from the domain of a rational expression?
All numbers that can cause the denominator to become zero should be placed next to the answer of the simplified rational expression, separated by a comma. As in 4/x-2, x<>2
What is the method for adding and subtracting rational expressions with different denominators?
a/b + c/d = (ad+bc)/bd , b<>0, d<>0
a/b - c/d = (ad-bc)/bd , b<>0, d<>0
How do you find the least common denominator of a rational expression?
1. Factor each denominator completely
2. List the factors of the first denominator
3. Add to the list in step 2 any factors of the second denominator that do not appear in the list
4. Form the product of each different factor from the list in step 3. This product is the least common denominator.
How do you add and subtract rational expressions that have different denominators with shared factors (Least common denominator)?
1. Find the least common denominator
2. Write all rational expressions in terms of the LCD. Such that you multiply both the numerator and the denominator of each rational expression by any factor(s) needed to convert the denominator into the LCD
3. Add or subtrack the numberators, placing the result over the LCD
4. If necessary, simplify the resulting rational expression
What are 2 methods of simplifying complex rational expressions?
1. Combine its numerator into a single expression and combine its denominator into a single expression. Then perform the division by inverting the denominator and multiplying
2. Find the LSD of all rational expression in the numerator and denominator. Multiply each term in its numerator and denominator by this LCD
What is an ordered pair?
In a rectangular coordinate system it is the point plotted at (x,y).
What is an Indentity Equation?
An equation that is true for all real number for which both sides are defined.
What is a conditional equations?
An equation that is not an identity equation but that is true for at least one real number.
What is an Inconsistent Equation?
An equation that is not true for even one real number.
What is the formula for simple interest?
I = Pr
I is the simple interest (annual)
P is the principal
r is the simple interest rate
What is the strategy for Problem Solving?
1. Let x represent one of the quantities.
2. Represent other quantities in terms of x
3. Write an equation in x that describes the conditions
4. Solve the equation and answer the question
5. Check the proposed solution in the original working of the problem
What is Mathematical Modeling?
The process of finding equations and formulas to describe real-world phenomena.
What is the strategy for Solving for a variable that occurs twice in a formula?
Consider the formula, Factor the duplicated variable on one side of the equation. Divide both sides by the resulting factor. Simplify.
A = P + Prt
A = P (1 + rt)
A/(1 + rt) = P

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