# Algebra 2 Quick Reference Sheet

Polynomials Completing the Square and Quadratics
 Definitions Monomial: a variable, a real number or the product of a real number and variables raised to a whole positive power Polynomial: an expression which is the sum of one or more monomials Root: the value of the variable when the polynomial is zero; also considered the solution to the polynomial function Degree: the highest exponent of a polynomial function Fundamental Theorem of Algebra: every polynomial equation with a degree greater the nzero has at least one root in the set of complex numbers Dividing Polynomials Divide by (x-5) Synthetic Division Find the root of the divisor. List all coefficients of the polynomial. 5| 1 -3 -5 -25 Bring down the first coefficient. Multiply the first coefficient by the root. Add the product to the second coefficient. Repeat the previous two steps for all The final integer is the remainder. Insert Variables starting with one less degree for each coefficient.  Long Division Divide the first term dividend by the first term divisor and distribute the result. Subtract the result from the dividend. Bring down the next term from the dividend. Repeat the previous 3 steps for each term. Finding Roots Factoring Polynomial Functions: a process where a polynomial is written as the multiplication of two or more different polynomials or monomials Complex Conjugates Theorem: If a+bi is a root of a polynomial function with real coefficients, then a-bi is also a root of the function Rational Root Theorem: Let f(x)=a polynomial function in standard form with integer coefficients; If p is all factors of the constant term and q is all factors of the leading coefficient, then (p/q) is all possible roots of y=f(x) Formulas for Factoring Greatest Common Factor Sum of Two Cubes Grouping Difference of Two Cubes General Trinomial Difference of Two Squares Perfect Squares  Completing the Square Quadratic Equation Quadratic Formula  two real roots a real, repeated root two complex root

Relations and Functions
Definitions

Relation: any set fo ordered pairs

Function: a relation which pairs each element of the domain with exactly one element of the range

Types of Functions

Algebraic Function: a function for which x is constant

Rational Functions: a function in which a polynomial function is divided by another polynomial function not equal to zero

Piecewise Functions: a function that is defined by different equations for different portions of the domain

Composite Functions: a function in which the variable is another function

Eamples
For the examples below,
f(x)=h and g(x)=k

Composite Functions

f(g(x))=f(x)

f(x)+g(x)=h+k

Subtracting Functions

f(x)-g(x)=h-k

Multiplying Functions Dividing Functions
Where k is not equal to zero Definitions

Natural Base: the irrational number, e, that is approximately 2.71828…

Logarithm: the inverse of an exponential function

Equivalent Exponent and Logarithmic Forms: For any positive base b, where b is greater than 0 and not equal to 1:
bx = y if and only if .
Common Log: the function ; can be shortened to Natural Log: the function in which the base, e, is the special number 2.71828…; function is also written as Formulas and Properties
Products of Exponents Quotient of Exponents  Negative Exponents Inverse Properties  Exponent of a Product Fractional Exponents Changing Bases Exponent of an Exponent Exponent of a Quotient      