Mathematics Fundamentals
Exponents
Overview
Raising a base number x to the power of n => multiplying the base number x by itself n times
Shorthand for multiplying the base number x by itself n times: x Superscript n question-mark Baseline equals y
Exponent (n) notation: x Superscript n
Operations
Procedure to MULTIPLY numbers with exponents, given they have the same base => ADD the exponents
Multiplication example: x squared times x Superscript 5 Baseline equals x Superscript 2 plus 5 Baseline equals x Superscript 7
Procedure to DIVIDE numbers with exponents, given they have the same base => SUBTRACT the exponents
Division example: StartFraction x squared Over x cubed EndFraction equals x Superscript 2 minus 3 Baseline equals x Superscript negative 1
Examples
Exponent example: 3 squared equals 3 times 3 equals 9
Radicals / Roots
Overview
Opposite of exponent - finding the base number x, given y (the result) and the exponent n of the base number - e.g. square root, cube root
To find the base number x: x question-mark Superscript n Baseline equals y
Notations
nth Root notation: RootIndex n StartRoot y EndRoot
Square Root notation: StartRoot y EndRoot
Cube Root notation: RootIndex 3 StartRoot y EndRoot
Examples
Square Root example: x squared equals 9 x equals StartRoot 9 EndRoot equals 3
Logarithms
Overview
The exponent n for a given base number x and result y
To find the exponent n to determine how many times to multiply a base number by itself to get the given result: x Superscript n question-mark Baseline equals y
Log notation: n equals log Subscript x Baseline left-parenthesis y right-parenthesis
Examples
Log example: 5 Superscript n question-mark Baseline equals 25 n equals log Subscript 5 Baseline left-parenthesis 25 right-parenthesis n equals 2
Common Logarithm
The "common logarithm" of a number is an exponential for the base 10
"Common Log notation - default of base 10: log left-parenthesis y right-parenthesis equals log Subscript 10 Baseline left-parenthesis y right-parenthesis
Log example: 10 Superscript n Baseline equals 1000 n equals log Subscript 10 Baseline left-parenthesis 1000 right-parenthesis equals log left-parenthesis 1000 right-parenthesis n equals 3
Natural Logarithm
The "natural logarithm (ln)" of a number is an exponential for the base e, where e is a constant ~2.718
Natural Log notation: ln left-parenthesis y right-parenthesis equals log left-parenthesis e right-parenthesis
Natural Log example: log Subscript e Baseline left-parenthesis 64 right-parenthesis equals ln left-parenthesis 64 right-parenthesis equals 4.1589
Inverse of Natural Log: e Superscript x
Derivative of ln x: StartFraction 1 Over x EndFraction
Properties / Laws
Distributive Property
Allows algebraic expressions in the form of a(b + c) to be reformulated to an equivalent expression
Distributive law: a left-parenthesis b plus c right-parenthesis equals a b plus a c
Khan Academy: Distributive Property
Monomials
Monomial - an algebric expression that contains only one term
Trinomials
Trinomial - an algebric expression that contains only three terms
Factoring
Rewriting a number or term as a product of several smaller factors or values in common
Difference of Squares Pattern
Every polynomial that is a difference of square terms can be factored with this formula: a squared minus b squared equals left-parenthesis a plus b right-parenthesis left-parenthesis a minus b right-parenthesis
Khan Academy: Factoring
Binomials
Overview
Binomial - an algebric expression that contains only two terms
Operations
Difference of Squares Pattern - special products of binomials (applying the Distributive Property twice): left-parenthesis a plus b right-parenthesis left-parenthesis a minus b right-parenthesis equals a left-parenthesis a plus b right-parenthesis minus b left-parenthesis a plus b right-parenthesis equals a squared plus a b minus a b minus b squared equals a squared minus b squared
Squaring binonmials of the form (a+b)^2: left-parenthesis a plus b right-parenthesis squared equals left-parenthesis a plus b right-parenthesis left-parenthesis a plus b right-parenthesis equals a squared plus 2 a b plus b squared
Squaring binonmials of the form (a-b)^2: left-parenthesis a minus b right-parenthesis squared equals left-parenthesis a minus b right-parenthesis left-parenthesis a minus b right-parenthesis equals a squared minus 2 a b plus b squared
Squaring binonmials of the form (ax+b)^2: left-parenthesis x plus a right-parenthesis squared equals left-parenthesis x plus a right-parenthesis left-parenthesis x plus a right-parenthesis equals x squared plus 2 a x plus a squared
Polynomials
Overview
Polynomial - an algebric expression that contains many terms - constants, variables with coefficients, and exponentials with a positive exponent
Notations
Standard Format - sorted by highest exponent variables first, then non-exponential variables (x to the power of 1), then constants: a Subscript n Baseline x Superscript n Baseline plus a Subscript n minus 1 Baseline x Superscript n minus 1 Baseline plus period period period plus a 2 x squared plus a 1 x plus a 0
Summation notation: sigma-summation Underscript k equals 0 Overscript n Endscripts a Subscript k Baseline x Superscript k
Examples
Rearranging into standard format (note the negative!): 6 plus 3 x minus 2 x squared equals minus 2 x squared plus 3 x plus 6
Operations
Addition/Subtraction of polynomials - add or subtract the coefficients of the like terms
Addition example: left-parenthesis 2 x squared plus 3 x plus 4 right-parenthesis plus left-parenthesis x squared minus 2 x plus 3 right-parenthesis equals 3 x squared plus x plus 7
Multiplication - multiply each term in the first expression with each term in the second expression
Multiplication Example: left-parenthesis 2 x squared plus 3 x plus 4 right-parenthesis times left-parenthesis x squared minus 2 x plus 3 right-parenthesis equals 2 x Superscript 4 Baseline minus x cubed plus 4 x squared plus x plus 12
Monomial Division (by a single term expression) - convert each term to a fraction and simplify
Divison Example: left-parenthesis 4 x squared plus 8 x right-parenthesis division-sign 2 x equals StartFraction 4 x squared Over 2 x EndFraction plus StartFraction 8 x Over 2 x EndFraction equals 2 x plus 4
Division by more than one term - use long division
Wikipedia: Polynomial Long Division
Khan Academy: Intro to long division of polynomials
Quadratic Equations
Overview
Graphs as a parabola. Vertex on top (A shaped) if the a in ax^2 is negative. Vertex on bottom (U shaped) if the a in ax^2 is positive
For every y value, there are two x values, equidistant from the line of symmetry
Line of symmetry x coordinate of vertex: x equals StartFraction negative b Over 2 a EndFraction
Notation for a quadratic equation: y equals a x squared plus b x plus c