Chapter 1

Foundations

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Summary

How to find the prime factorization of a composite number. Find two factors whose product is the given number, and use these numbers to create two branches If a factor is not prime, write it as the product of two factors and continue the process Divisibility Tests A number is divisible by: 2 if the last digit is 0, 2, 4, 6, or 8.

Key terms

composite number

A composite number is a counting number that is not prime. It has factors other than 1 and the number itself

prime factorization

The prime factorization of a number is the product of prime numbers that equals the number

number

divisible by: 2 if the last digit is 0, 2, 4, 6, or 8

factors

If a · b = m , then a and b are factors of m

factor

prime, that branch is complete

denominator

In a fraction, written a b , where b ≠ 0 , the denominator b is the number of equal parts the whole has been divided into

fraction

A fraction is written a b , where b ≠ 0 , and a is the numerator and b is the denominator. A fraction represents parts of a whole

Chapter 2

Solving Linear Equations

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Summary

How to determine whether a number is a solution to an equation Step 1. Substitute the number in for the variable in the equation If it is true, the number is a solution. If it is not true, the number is not a solution

Key terms

linear equation

A linear equation is an equation in one variable that can be written, where a and b are real numbers and a ≠ 0 , as a x + b = 0

number

a solution to an equation Step 1

conditional equation

An equation that is true for one or more values of the variable and false for all other values of the variable is a conditional equation

identity

An equation that is true for any value of the variable is called an Identity. The solution of an identity is all real numbers

contradiction

An equation that is false for all values of the variable is called a contradiction. A contradiction has no solution

solution of an equation

A solution of an equation is a value of a variable that makes a true statement when substituted into the equation

compound inequality

A compound inequality is made up of two inequalities connected by the word “and” or the word “or.”

Chapter 3

Graphs and Functions

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Summary

Graph of a Linear Equation: The graph of a linear equation A x + B y = C is a straight line. How to graph a linear equation by plotting points. Points with an x -coordinate equal to 0 are on the y -axis, and have coordinates ( 0 , b ) Every point on the line is a solution of the equation.

Key terms

function

A function is a relation that assigns to each element in its domain exactly one element in the range

linear equation

An equation of the form A x + B y = C , where A and B are not both zero, is called a linear equation in two variables

x -coordinate equal to 0

on the y -axis, and have coordinates ( 0 , b )

line

a solution of the equation

boundary line

The line with equation A x + B y = C is the boundary line that separates the region where A x + B y > C from the region where A x + B y < C

linear inequality

A linear inequality is an inequality that can be written in one of the following forms: A x + B y > C , A x + B y ≥ C , A x + B y < C , A x + B y > C , A x + B y ≥ C , A x + B y…

mapping

A mapping is sometimes used to show a relation. The arrows show the pairing of the elements of the domain with the elements of the range

ordered pair

An ordered pair, ( x , y ) gives the coordinates of a point in a rectangular coordinate system. The first number is the x -coordinate. The second number is the y -coordinate

Chapter 4

Systems of Linear Equations

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Summary

How to solve a system of linear equations by graphing. Determine whether the lines intersect, are parallel, or are the same line If the lines intersect, identify the point of intersection. If the lines are parallel, the system has no solution.

Key terms

system of linear equations

When two or more linear equations are grouped together, they form a system of linear equations

lines

parallel, the system has no solution

consistent and inconsistent systems

Consistent system of equations is a system of equations with at least one solution; inconsistent system of equations is a system of equations with no solution

cost function

The cost function is the cost to manufacture each unit times x , the number of units manufactured, plus the fixed costs; C ( x ) = (cost per unit) x + fixed costs

row-echelon form

A matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros

Chapter 5

Polynomials and Polynomial Functions

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Summary

Degree of a Polynomial The degree of a term is the sum of the exponents of its variables A monomial in one variable is a term of the form a x m , where a is a constant and m is a whole number Polynomials Polynomial —A monomial, or two or more algebraic terms combined by addition or subtraction is a polynomial Monomial —A polynomial with exactly one term is called a monomial

Key terms

polynomial

A monomial or two or more monomials combined by addition or subtraction is a polynomial

polynomial function

A polynomial function is a function whose range values are defined by a polynomial

monomial

A monomial is an algebraic expression with one term. A monomial in one variable is a term of the form a x m , where a is a constant and m is a whole number

monomial in one variable

a term of the form a x m , where a is a constant and m is a whole number

degree of a polynomial

The degree of a polynomial is the highest degree of all its terms

degree of a term

The degree of a term is the sum of the exponents of its variables

trinomial

A trinomial is a polynomial with exactly three terms

binomial

A binomial is a polynomial with exactly two terms

Chapter 6

Factoring

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Summary

How to find the greatest common factor (GCF) of two expressions. Write all variables with exponents in expanded form

Key terms

factoring

Splitting a product into factors is called factoring

greatest common factor

The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions

Zero Product Property

The Zero Product Property says that if the product of two quantities is zero, then at least one of the quantities is zero

polynomial equation

A polynomial equation is an equation that contains a polynomial expression

zero of the function

A value of x where the function is 0, is called a zero of the function

degree of the polynomial equation

The degree of the polynomial equation is the degree of the polynomial

quadratic equation

Polynomial equations of degree two are called quadratic equations

Chapter 7

Rational Expressions and Functions

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Summary

Opposites in a Rational Expression The opposite of a - b is b - a . Multiplication of Rational Expressions If p , q , r , and s are polynomials where q ≠ 0 , s ≠ 0 , then p q · r s = p r q s

Key terms

rational expression

A rational expression is an expression of the form p q , where p and q are polynomials and q ≠ 0

complex rational expression

A complex rational expression is a rational expression in which the numerator and/or denominator contains a rational expression

rational function

A rational function is a function of the form R ( x ) = p ( x ) q ( x ) where p ( x ) and q ( x ) are polynomial functions and q ( x ) is not zero

similar figures

Two figures are similar if the measures of their corresponding angles are equal and their corresponding sides have the same ratio

proportion

When two rational expressions are equal, the equation relating them is called a proportion

rational inequality

A rational inequality is an inequality that contains a rational expression

Chapter 8

Roots and Radicals

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Summary

Properties of a n When n is an even number and a ≥ 0 , then a n is a real number Square Root Notation m is read ‘the square root of m ’ The square root of m , is a positive number whose square is m N th Root of a Number If b n = a , then b is an n th root of a

Key terms

n When n

an even number and a ≥ 0 , then a n is a real number

complex number

A complex number is of the form a + bi , where a and b are real numbers. We call a the real part and b the imaginary part

rationalizing the denominator

Rationalizing the denominator is the process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer

radical equation

An equation in which a variable is in the radicand of a radical expression is called a radical equation

standard form

A complex number is in standard form when written as a + b i , where a, b are real numbers

complex number system

The complex number system is made up of both the real numbers and the imaginary numbers

like radicals

Like radicals are radical expressions with the same index and the same radicand

imaginary unit

The imaginary unit i is the number whose square is -1. i 2 = -1 or i = -1

Chapter 9

Quadratic Equations and Functions

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Summary

Binomial Squares Pattern If a and b are real numbers, Square Root Property If x 2 = k , then x = k or x = - k or x = ± k Isolate the quadratic term and make its coefficient one Find ( 1 2 b ) 2 , the number to complete the square

Key terms

quadratic function

A quadratic function, where a , b , and c are real numbers and a ≠ 0 , is a function of the form f ( x ) = a x 2 + b x + c

discriminant

In the Quadratic Formula, x = - b ± b 2 - 4 a c 2 a , the quantity b 2 - 4 ac is called the discriminant

quadratic inequality

A quadratic inequality is an inequality that contains a quadratic expression

Chapter 10

Exponential and Logarithmic Functions

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Summary

Inverse of a Function Defined by Ordered Pairs: If f ( x ) is a one-to-one function whose ordered pairs are of the form ( x , y ) , then its inverse function f -1 ( x ) is the set of ordered pairs ( y , x ) Horizontal Line Test: If every horizontal line, intersects the graph of a function in at most one point, it is a one-to-one function Inverse Functions: For every x in the domain of one-to-one function f and f -1 , f -1 ( f ( x ) = x f ( f -1 ( x ) = x f -1 ( f ( x ) = x f ( f -1 ( x ) = x How to Find the Inverse of a One-to-One Function: Step 1.

Key terms

logarithmic function

The function f ( x ) = log a x is the logarithmic function with base a , where a > 0 , x > 0 , and a ≠ 1 . y = log a x is equivalent to x = a y

one-to-one function

A function is one-to-one if each value in the range has exactly one element in the domain. For each ordered pair in the function, each y -value is matched with only one x -value

exponential function

An exponential function, where a > 0 and a ≠ 1 , is a function of the form f ( x ) = a x

asymptote

A line which a graph of a function approaches closely but never touches

common logarithmic function

The function f ( x ) = log x is the common logarithmic function with base 10 , where x > 0 . y = log x is equivalent to x = 10 y

natural base

The number e is defined as the value of ( 1 + 1 n ) n , as n gets larger and larger. We say, as n increases without bound, e ≈ 2.718281828

natural exponential function

The natural exponential function is an exponential function whose base is e : f ( x ) = e x . The domain is ( - ∞ , ∞ ) and the range is ( 0 , ∞ )

Chapter 11

Conics

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Summary

Circle: A circle is all points in a plane that are a fixed distance from a fixed point in the plane. The given point is called the center, ( h , k ) , and the fixed distance is called the radius, r, of the circle Parabola: A parabola is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.

Key terms

parabola

A parabola is all points in a plane that are the same distance from a fixed point and a fixed line

given point

called the center, ( h , k ) , and the fixed distance is called the radius, r, of the circle

circle

A circle is all points in a plane that are a fixed distance from a fixed point in the plane

fixed point

called the focus, and the fixed line is called the directrix of the parabola

hyperbola

A hyperbola is defined as all points in a plane where the difference of their distances from two fixed points is constant

ellipse

An ellipse is all points in a plane where the sum of the distances from two fixed points is constant

system of nonlinear equations

A system of nonlinear equations is a system where at least one of the equations is not linear

Chapter 12

Sequences, Series and Binomial Theorem

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Summary

General Term ( n th term) of an Arithmetic Sequence The general term of an arithmetic sequence with first term a 1 and the common difference d is a n = a 1 + ( n - 1 ) d Sum of the First n Terms of an Arithmetic Sequence The sum, S n , of the first n terms of an arithmetic sequence, where a 1 is the first term and a n is the n th term is S n = n 2 ( a 1 + a n ) General Term ( n th term) of a Geometric Sequence: The general term of a geometric sequence with first term a 1 and the common ratio r is a n = a 1 r n - 1 Sum of the First n Terms of a Geometric Series: The sum, S n , of the n terms of a geometric sequence is S n = a 1 ( 1 - r n ) 1 - r where a 1 is the first term and r is the common ratio

Key terms

sequence

A sequence is a function whose domain is the counting numbers

common difference

The difference between consecutive terms in an arithmetic sequence, a n - 1 , is d , the common difference, for n greater than or equal to two

common ratio

The ratio between consecutive terms in a geometric sequence, a n - 1 , is r , the common ratio, where n is greater than or equal to two

arithmetic sequence

An arithmetic sequence is a sequence where the difference between consecutive terms is constant

geometric sequence

A geometric sequence is a sequence where the ratio between consecutive terms is always the same

infinite geometric series

An infinite geometric series is an infinite sum infinite geometric sequence

first term and a n

the n th term is S n = n 2 ( a 1 + a n )