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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 $(x^3-3x^2-5x-25)$ by (x-5)
Synthetic Division Find the root of the divisor. $(x-5)=0 \rightarrow x=5$ 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. $x^2+2x+5$ $R=\frac{0}{x-5}$	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 $a^3b+a^2b^2=ab(a^2+ab+b)$	Sum of Two Cubes $a^3+b^3=(a+b)(a^2-ab+b^2)$
Grouping $ax+cx+ay=(a+c)(x+y)$	Difference of Two Cubes $a^3-b^3=(a-b)(a^2+ab+b^2)$
General Trinomial $acx^2+adx+bcx+bd=(ax+b)(cx+d)$	Difference of Two Squares $a^2-b^2=(a+b)(a-b)$
Perfect	Squares
$a^2+2ab+b^2=(a+b)^2$	$a^2-2ab+b^2=(a-b)^2$

Completing the Square
$(ax)^2+abx+(\frac{b}{2})^2 = (ax+\frac{b}{2})^2$

Quadratic Equation
$ax^2+bx+c=0$

Quadratic Formula $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

$b^2-4ac>0 \rightarrow$ two real roots

$b^2-4ac=0 \rightarrow$ a real, repeated root

$b^2-4ac<0 \rightarrow$ 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)

Adding Functions

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

Subtracting Functions

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

Multiplying Functions

$f(x)\cdot g(x)=hk$

Dividing Functions
Where k is not equal to zero $\frac{f(x)}{g(x)}=\frac{h}{k}$

Logarithms, Exponents,

Radicals

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 $x = \log_b{(y)}$ .

Common Log: the function $f(x)=\log_{10}{x}$ ; can be shortened to $f(x)=\log{x}$

Natural Log: the function $f(x)=\log_e{x}$ in which the base, e, is the special number 2.71828…; function is also written as $f(x)=\ln{x}$

Formulas and

Properties

Products of Exponents
$b^m \cdot b^n = b^{m+n}$

Quotient of Exponents
$\frac{a^m}{a^n}=a^{m-n}$

Quotient of Radicals
$\sqrt[n]{\frac{a}{b}}$

Negative Exponents
$x^{-a}=\frac{1}{x^a}$

Inverse Properties
$\log_b{b^x}=x$
$\b^{\log_b{x}=x$

Exponent of a Product
$(ab)^m=a^m b^m$

Fractional Exponents
$x^{\frac{a}{b}}=\sqrt[b]{x^a}=(\sqrt[b]{x})^a$

Changing Bases
$\log_b{x}=\frac{\log_a{x}}{\log_a{x}}$

Exponent of an Exponent
$(a^m)^n=a^{m\cdot n}$

Exponent of a Quotient
$(\frac{a}{b})^m=\frac{a^m}{b^m}$
$(\frac{a}{b})^{-n}=(\frac{b}{a})^n=\frac{b^n}{a^n}$

Product of Radicals
$\sqrt[n]{ab}=\sqrt[n]{a} \cdot \sqrt[n]{n}$

Product of Logarithms
$\log_b{mn}=\log_b{m}+\log_b{n}$

Quotient of Logarithms
$\log_b{(\frac{m}{n})}=log_b{m}-log_b{n}$

Powers of Logarithms
$\log_b{(m^p)}=p \cdot \log_b{m}$

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Imaginary and Complex Numbers

Series and Sequences

Definitions
Imaginary Number: the number $i$ that squared yields a result of -1.
Complex Number: a number that can be written in the form $a+bi$ where the $a$ term is real and the $bi$ term is imaginary
Powers of i
To find $i^n$ , divide n by 4 and match the remainder to one of the powers of $i$ to the right. $i=\sqrt{-1}$ $i^2=-1$ $i^3=-i$ $i^4=1$

Definitions
Sequence: an ordered list of numbers or terms
Explicit Formula: a formula that defines all the terms of a sequence
Recursive Formula: a formula that uses one or more previous terms to generate the next term
Series: an expression that indicates the sum of terms of a sequence
Arithmetic Sequence: a sequence whose successive terms differ by the same common difference, d
Arithmetic Series: the indicated sum of the terms of an arithmetic sequence
Geometric Sequence: a sequence in which the ratio of successive terms is the same common ratio, r
Geometric Series: the indicated sum of the terms of a geometric sequence
Binomial Theorem: a formula for finding the expansion of the powers of a binomial expression

Conic Sections

Formulas

Definitions
Circle: the set of all points in a plane that are equidistant from a given point, the center in the plane.
Ellipse: the set of all points in a plane such that the sum of the distances from two fixed points, the foci, is constant
Parabolas: the set of all points in a plane that are the same distance from a given point, the focus, and a given line, the directrix
Hyperbola: the set of all points in a plane such that the absolute value of the difference of the distances from two fixed points, the foci, is constant
Standard Equations
(h,k)= center or vertex r=radius of circle

Circle

$(x-h)^2+(y-k)^2=r^2$

Ellipse For a>b	Horizontal Major Axis $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$	Vertical Major Axis $\frac{(y-k)^2}{a^2}+\frac{(x-h)^2}{b^2}=1$
Parabola	Open up if a>0, down if a<0 $y=a(x-h)^2+l$	Opens right if a>0 left if a<0 x=a(y-k)^2+h
Hyperbola	Horizontal Transverse Axis $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$	Vertical Transverse Axis $\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$

Arithmetic Sequence $a_n=a_1+(n-1)d$	Geometric Sequence $a_n=a_1r^{n-1}$
Arithmetic Series $S_n=\frac{n}{2}(a_1+a_n)$	Geometric Series $S_n=\frac{a_1(1-r^n)}{1-r}$

Binomial Theorem
${(a+b)}^n = {\sum\limits_{k=0}^n{n\choose{k}} \cdot a^{n-k}\cdot{b^k}}$ ; ${n\choose{k}}=\frac{n!}{(n-k)!k!}$

Probability

Other Notes
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Definitions
Fundamental Counting Principle: if an event can occur in m ways and another can occur in n ways, then the first event followed by the second event can occur in $m\cdot n$ ways
Independent Events: Events that do not affect the outcome of each other
Dependent Events: Events that affect he outcomes of each other

Mutually Exclusive Events: two elements that cannot occur at the same time
Inclusive Events: Two events whos outcomes may be the same
Combination: An arrangement of objects in which order is not important
Permutation: An arrangement of objects in which order is important
Formulas

Probability with Two Independent Events $P({A}\cap{B})=P(A) \cdot P(B)$	Probability with Two Dependent Events $P({A}\cap{B})=P(A) \cdot P(A\|B)$	Combination $C(n,r)=\dfrac{n!}{(n-r)!r!}$
Probability of Success $P(s)=\dfrac{s}{(s+f)}$	Probability of Mutually Exclusive Events $P(s)=\dfrac{s}{(s+f)}$	Permutation $P(n,r)=\dfrac{n!}{(n-r)!}$

Probability of Inclusive Events
$P({A}\cup{B})=P(A)+P(B)-P({A}\cap{B})$

Permutation with Repetitions $P(n)=\dfrac{n!}{p!q!}$

Matrices and Determinates

Definitions
Matrix: A rectangular array of variables or constants in horizontal rows and vertical columns, usually enclosed in brackets.
Adding/Subtracting Matrices: addition or subtraction of matrices may only occur if the matrices are the same dimensions; corresponding entries are added or subtracted to form the result
Multiplying Matrices by a Constant: each entry of the matrix is multiplied by the indicated constant
Multiplying Matrices by another Matrix: the number of columns in the first matrix must be the same as the number of rows in the second matrix; multiply each
Determinate: a square array of numbers or variables enclosed between two parallel lines.

Formulas

Addition and Subtraction of Matrices $\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \pm \begin{bmatrix}j & k & l\\ m & n & o\\ p & q & r\end{bmatrix} = \begin{bmatrix} a \pm j & b \pm k & c \pm l\\ d \pm m & e \pm n & f \pm o\\ g \pm p & h \pm q & i \pm r \end{bmatrix}$	2nd Order Determinate $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad-bc$
Multiplying MatricesBy a constant, k k $\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} ka & kb \\kc & kd \end{bmatrix}$ By another matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix} \cdot \begin{bmatrix} e & f \\ g & h \end{bmatrix} = \begin{bmatrix} ae+bg & af+bh \\ ce+dg & cf+dh \end{bmatrix}$	3rd Order Determinate $\begin{vmatrix} a & b & c \\ d & e & f\\ g & h & i \end{vmatrix}=a \begin{vmatrix} e & f \\ h & i \end{vmatrix} - b \begin{vmatrix} d & f \\ g & i \end{vmatrix} +c \begin{vmatrix} d & e \\ g & h \end{vmatrix}$

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