Algebra / Reference

Algebra Cheat Sheet

Every rule, formula, and pattern from the Algebra lessons on this site, collected on one page — exponents, radicals, polynomials, factoring, rational expressions, equations and inequalities, the quadratic formula, lines, functions and their graphs, conic sections, polynomial theorems, exponentials and logarithms, plus the mistakes that cost the most points and quick checks that catch them. Skim it before a test, or download the PDF and keep a copy on your desk.

Download Cheat Sheet (PDF)

Arithmetic and Fraction Rules

Rule	Statement
Order of operations	Parentheses, then exponents, then multiply/divide left to right, then add/subtract left to right
Distribute a factor
Multiply a fraction by a number
Fraction divided by a number
Number divided by a fraction
Adding unlike fractions
Subtracting unlike fractions
Dividing two fractions
Canceling a true common factor	,

Exponent Rules

Rule	Statement
Product Rule
Quotient Rule
Power Rule
Power of a Product
Power of a Quotient
Zero Exponent	,
Negative Exponent
Negative Exponent of a Quotient
Rational Exponent

Full lessons: Exponent rules · Rational exponents

Radical Rules

Rule	Statement
Root as a fractional exponent
Product Rule
Quotient Rule
Root of a root
Odd-index root of a power	( odd)
Even-index root of a power	( even)
Simplify a square root	Pull out the largest perfect-square factor:
Rationalize a single root
Rationalize with a conjugate

Full lessons: Simplifying radicals · Rationalizing denominators

Polynomials and Special Products

Rule	Statement
Degree and leading coefficient	In with : degree , leading coefficient
Adding and subtracting	Combine like terms; a minus sign in front of brackets flips every sign inside:
FOIL (two binomials)
Square of a sum
Square of a difference
Sum times difference
Cube of a sum
Cube of a difference

Full lessons: Polynomials · Multiplying polynomials

Factoring Formulas

Pattern	Factors as
Greatest common factor (always first)
Grouping (four terms)
Difference of squares
Perfect square (sum)
Perfect square (difference)
General trinomial
Trinomial (ac method)	Find with and ; split , then factor by grouping
Sum of cubes
Difference of cubes
Difference of even powers
Sum of squares	does not factor over the real numbers

Full lessons: Factoring polynomials · Factoring quadratics

Rational Expressions

Operation	Rule
Domain	Exclude every value that makes a denominator
Simplify	Factor fully, then cancel common factors only: ,
Multiply
Divide
Add or subtract	Rewrite over the least common denominator, then combine numerators:
Complex fraction	Multiply the top and bottom by the LCD of every small fraction inside it

Full lessons: Rational expressions · Simplifying rational expressions

Complex Numbers

Rule	Statement
Imaginary unit	,
Powers of	, , , ; the cycle repeats every four powers
Square root of a negative	,
Add / subtract	Combine real parts and imaginary parts separately
Multiply	FOIL, then replace every with
Conjugate	The conjugate of is
Conjugate product
Modulus
Divide	Multiply the numerator and denominator by the denominator’s conjugate

Full lessons: Complex numbers · Complex number arithmetic

Linear Equations and Word Problems

Situation	Formula or method
Solving a linear equation	Clear fractions (multiply by the LCD), distribute, collect the -terms on one side, divide by the coefficient
Solving for a variable	Treat every other letter as a number:
Distance, rate and time
Work done together
Mixture	, where conc = concentration
Simple interest
Percent

Full lessons: Linear equations · Word problems

Inequalities and Absolute Value

Rule	Statement
Add or subtract the same value	If , then and
Multiply or divide by a positive	If and , then
Multiply or divide by a negative	If and , then — direction flips
Compound "and"	means and at the same time — the overlap
Compound "or"	Keep every that satisfies either part — the union
Absolute value, by definition	if ; if
Product and quotient	,
Triangle inequality
Solving ()	or
Solving ()
Solving ()	or
Negative right side	and have no solution; holds for every

Full lessons: Linear inequalities · Absolute value inequalities

Interval Notation

Inequality	Interval






or
All real numbers
Brackets	Square bracket = endpoint included; round bracket = excluded; always takes a round bracket

Full lesson: Interval notation

Solving Quadratics

Method	Statement
Zero Product Property	If , then or — factor, then set each factor equal to
Quadratic Formula	For ():
Discriminant
Discriminant	Two distinct real solutions
Discriminant	One repeated real solution
Discriminant	Two complex solutions
Square Root Property	If , then
Completing the square

Completing the square, worked example. Solve .

Divide every term by the leading coefficient :
Move the constant to the other side:
Halve the -coefficient, square it, add to both sides:
Factor the left side:
Apply the Square Root Property:
Solve for : , so or

Full lessons: Quadratic formula · Completing the square

Other Equations and Inequalities

Type	Method
Equation with fractions	Multiply every term by the LCD; then reject any answer that makes a denominator
Radical equation	Isolate the radical, raise both sides to the index, solve, then check every answer — squaring can create extraneous roots
Quadratic form	Substitute for the repeated piece: becomes with
Absolute value equation	Isolate first, then split into or
Polynomial inequality	Move everything to one side, find the zeros, then test one value in each interval (sign chart)
Rational inequality	Zeros of the numerator and of the denominator split the number line; denominator zeros are never included

Full lessons: Equations with radicals · Polynomial inequalities

Lines and Function Forms

Form	Equation
Slope of a line through two points
Slope-intercept form	, -intercept
Point-slope form
Standard form	, slope
Horizontal and vertical lines	has slope ; has undefined slope
Parallel lines	Equal slopes:
Perpendicular lines	, so
Intercepts	x-intercept: set ; y-intercept: set
Parabola, vertex form	, vertex
Parabola, standard form	, vertex at
Distance between two points
Midpoint of two points

Full lessons: Lines · Perpendicular lines

Functions

Idea	Rule
Function	Every input has exactly one output; its graph passes the vertical line test
Domain restrictions	Denominator ; even-root radicand ; logarithm argument
Combining functions	, , with
Composition	— apply first
One-to-one	; the graph passes the horizontal line test
Finding an inverse	Swap and , solve for ; check that
Graph of an inverse	Reflection of across ; domain and range swap
Even function	— symmetric about the y-axis
Odd function	— symmetric about the origin
Piecewise function	Use the rule whose interval contains the input

Full lessons: Functions · Inverse functions

Parent Functions

Function	Domain · Range · Key feature
Constant	· · horizontal line
Linear	· · slope through the origin
Quadratic	· · vertex
Cubic	· · symmetric about the origin
Absolute value	· · V shape, vertex
Square root	· · starts at
Reciprocal	all · all · asymptotes and
Exponential	· · through , asymptote
Logarithmic	· · through , asymptote

Full lesson: Graphing functions

Graph Transformations

Equation	Effect on the graph of
	Shift up (down if )
	Shift right (left if ) — inside the function, signs act backwards
	Reflect across the x-axis
	Reflect across the y-axis
	Vertical stretch by if ; compression if
	Horizontal compression by a factor of if ; stretch if
	Stretch or reflect first, then shift; the point of the parent moves to

Full lesson: Function transformations

Conic Sections

Shape	Equation
Circle	, center , radius
Ellipse
Ellipse foci	, with the larger semi-axis; foci lie on the major axis
Hyperbola (opens left/right)
Hyperbola (opens up/down)
Hyperbola asymptotes	for both forms above ( under the -term, under the -term)
Hyperbola foci
General form to standard form	Complete the square in and in separately, then divide to make the right side

Full lessons: Circles · Ellipses · Hyperbolas

Rational Functions and Asymptotes

Feature	How to find it
Vertical asymptote	Where the denominator is after common factors are canceled
Hole	A factor that cancels from the top and bottom leaves a hole, not an asymptote
Horizontal asymptote, top degree smaller
Horizontal asymptote, equal degrees
Horizontal asymptote, top degree larger	None
Slant asymptote	Top degree exactly one more than the bottom: divide, and the quotient is the asymptote

Full lessons: Rational functions · Horizontal asymptotes

Polynomial Functions

Theorem or rule	Statement
Division algorithm	, with
Remainder Theorem	Dividing by leaves the remainder
Factor Theorem	exactly when is a factor
Rational Root Theorem	Possible rational zeros are , a factor of , a factor of
Descartes’ Rule of Signs	Positive zeros = sign changes in , or fewer by an even number; use for negative zeros
Fundamental Theorem of Algebra	A degree- polynomial has exactly complex zeros, counting multiplicity
Conjugate pairs	With real coefficients, if is a zero then so is
Multiplicity	Odd multiplicity: the graph crosses the x-axis; even: it touches and turns back
Turning points	At most for degree
End behavior	Even degree: ends match; odd degree: ends opposite; positive leading coefficient: right end rises
Writing from zeros
Partial fractions	when

Full lessons: Finding polynomial zeros · End behavior

Exponential Functions

Rule	Statement
Exponential function	, , ; is the value at
Growth or decay	: growth; : decay
Percent rate	Growth ; decay
Compound interest
Continuous compounding	,
Half-life and doubling time	and
Equation from two points	, then
Graph of	Horizontal asymptote ; domain all reals; range above if

Full lessons: Exponential functions · Growth and decay

Logarithm Rules

Rule	Statement
Definition	means
Worked example	, because
Identity values	,
Inverse of an exponential	,
Power Rule
Product Rule
Quotient Rule
Change of base
One-to-one property
Natural and common log	;
Natural log with	, ,
Domain	is only defined for

Full lessons: Logarithm rules · Change of base formula

Solving Exponential and Logarithmic Equations

Situation	Method
Both sides share a base	: , so
No common base	Isolate the power, take of both sides:
Doubling time at rate
A log equal to a number
Several logs	Condense to a single log with the rules first, then remove it
Always check	Reject any answer that makes a log argument — it is extraneous

Full lessons: Solving exponential equations · Solving logarithmic equations

Common Algebra Mistakes

What goes wrong	Why, with an example
treated as or as	Division by zero is undefined — it isn’t any number at all.
	Without parentheses, the exponent binds to only: , while .
	A power of a power multiplies exponents: , not .
	Check with numbers: , but — not equal.
	Only cancels: ; the -term keeps its denominator.
	The negative sign has to distribute to both terms: .
	Expanding fully: — the middle term is real.
	, not — a square root doesn’t distribute over a sum.
	Squaring comes first: , while .
	Any nonzero base to the zero power is , not .
	It is : , not .
Dividing both sides by	It loses the solution : from , factor to keep both and .
Keeping the inequality sign after multiplying by a negative	gives , not .
	; exponents add only when the bases match.
	No rule splits the log of a sum: , but .
	A quotient of logs is a change of base, not the quotient rule: , but .

Tips and Quick Checks

Tip	Why it helps
Substitute your answer back in	The fastest way to catch arithmetic slips and the extraneous roots that radicals, logs and fractions produce
Factor out the GCF first	Every factoring step after it works with smaller numbers
Write down domain restrictions before solving	Denominators, even roots and logs tell you in advance which answers to reject
Estimate before you calculate	must lie between and , because and
Test a rule with numbers	Unsure whether a step is legal? Try it with , — one counterexample settles it
Keep exact values until the end	Round only the final answer; early rounding grows, especially inside powers
Check the discriminant first	With integer coefficients, a perfect-square discriminant means the quadratic factors over the integers; otherwise use the formula
Sketch the graph	Intercepts, end behavior and asymptotes expose sign errors that the algebra hides
Put negatives in brackets on a calculator	but
Answer the question that was asked	Interval notation, exact form or a rounded decimal — match the format the question requests

Every rule above gets a full derivation, worked examples, and practice problems in the Algebra lessons. Build speed on any of these topics with the Math Practice Problems.