Algebra 2 Quick Reference Sheet

Polynomials Completing the Square and Quadratics

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
Sum of Two Cubes
Difference of Two Cubes
General Trinomial
Difference of Two Squares
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

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

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

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

Composite Functions


Adding Functions


Subtracting Functions


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

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
Quotient of Radicals
Negative Exponents
Inverse Properties
Exponent of a Product
(ab)^m=a^m b^m
Fractional Exponents
Changing Bases
Exponent of an Exponent
(a^m)^n=a^{m\cdot n}
Exponent of a Quotient
Product of Radicals
\sqrt[n]{ab}=\sqrt[n]{a} \cdot \sqrt[n]{n}
Product of Logarithms
Quotient of Logarithms
Powers of Logarithms
\log_b{(m^p)}=p \cdot \log_b{m}

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