Transcript A Quick Review of MTH070

A Quick Review of
MTH070
Elementary Algebra
Algebraic Notation
Algebraic Properties & Simplifying Expressions
Linear Equations, Formulas, & Inequalities
Graphs & Equations of Lines
Systems of Equations
Functions
Operations with Polynomials
Copyright © 2010 by Ron Wallace, all rights reserved.
Algebraic Properties
• Identities: 0 & 1
• Inverses: Opposites & Reciprocals
Subtraction  Adding Opposites
Division  Multiplying Reciprocals
• Commutative
Order of addition & multiplication
• Associative
Multiple additions & multiplications
• Distributive
a(b + c) = ab + ac
Simplify Expressions
• Remove grouping symbols.
– Associative & Distributive Properties
• Combine like terms.
– Distributive Property
• Complete all possible rational arithmetic.
Solve Linear Equations
• Simplify both sides of the equation.
– Result: ax + b = cx + d
• Clear Fractions (Optional)
• Apply the addition principle & simplify
– Move all linear terms to one side and all other terms to
the other side.
– To move a term to the other side, change its sign.
– Result: ex = f
• Apply the multiplication principle & simplify
– Divide both sides by the coefficient of the linear term.
• Check
Formulas
• Apply
– Given values for all but one variable, determine the value
of the remaining variable.
• Solve for a specified variable in terms of the other
variables.
– Same procedure as linear equation w/ one variable.
– Treat all other variables like constants.
Solve Inequalities
• Same as solving equations with two exceptions
– When multiplying or dividing both sides of the inequality
by a negative number, the direction of the inequality is
reversed.
– When switching sides of an inequality, the direction of the
inequality is reversed.
• Graphing inequalities …
(
x>a
x<a
a
[
a
)
a
xa
xa
]
a
Rectangular Coordinates
• AKA …
– Cartesian
Coordinates
– The XY-plane
• Ordered pairs
(x, y)
• Lines
x-intercept: (a, 0)
y-intercept: (0, b)
Slope: m  rise
run 
y2  y1
x2  x1
Equations of Lines
• Standard Form:
Ax + By = C
x-intercept: (C/A, 0)
y-intercept: (0, C/B)
• Slope-Intercept Form: y = mx + b
y-intercept: (0, b)
Slope: m
• Point-Slope Form:
Point: (h, k)
Slope: m
• Horizontal Lines:
• Vertical Lines:
y – k = m(x – h)
y = k + m(x – h)
y=b
x=a
Graphing Lines
• Option 1:
A picture of ALL
of the solutions
of an equation.
y = mx + b
– Plot the y-intercept: (0, b)
– Find a second point using the slope (rise/run)
– Draw the line
• Option 2:
Ax + By = C
– Plot the x-intercept: (C/A, 0)
– Plot the y-intercept: (0, C/B)
– Draw the line
• Checking … find and verify another solution.
Solve Systems of Equations
• Graphing
– Graph both equations.
– The point of intersection is the solution.
• Substitution
– Solve one equation for one variable.
– Substitute into the other equation and solve.
– Use the result to determine the other variable.
• Elimination
– Multiply to get coefficients of a variable to be opposites.
– Add the equations and solve.
– Repeat for the other variable.
• Check (in both equations)
Functions
• f(x) = expression in x  y = expression in x
• Domain: Set of all values for x.
• Range: Set of all possible results.
• f(3)
– Replace all occurrences of x with 3
– Simplify.
• Linear Function:
f(x) = mx + b
Rules of Exponents
a a a
m
n
mn
m
a
mn
a
n
a
a 
a
mn
a    a
m
m
a b
m m
a
a
   m
b
b
m
m n
 ab 
m

m
a  1, a  0
0
(a)  (1) a
m
m
m
Polynomials - Terminology
• Term
– constant term
– linear term
– quadratic term
– leading term
– leading coefficient
• Coefficient
• Degree
– of a term
– of a polynomial
• Polynomial
– monomial
– binomial
– trinomial
• Evaluating a …
– polynomial
– polynomial function
Polynomials - Operations
• Addition
– Add like terms
• Subtraction
– Find the opposite of the polynomial being subtracted
– Add
• Multiplication
– Multiply each term of the first by each term of the second
– Combine like terms
• Division
– By monomial … divide each term (properties of exponents)
– By polynomial … long division
• Synthetic division: Only with division by x-a or x+a
Polynomials
Special
Products
• FOIL
• “similar” – O & I terms are like terms
• (a + b)(a – b) = a2 – b2
• (a + b)2 = a2 + 2ab + b2
• (a – b)2 = a2 – 2ab + b2