Download MATH-1310 Review Concepts (Haugen) Linear Equations

MATH-1310 Review Concepts (Haugen)
Unit 1 Linear Equations
Exam 1 – Sections 3.2, 3.3, 3.5, 4.1, 4.2, and 4.3
Linear Equations
Rise y2  y1

Run x2  x1
Find the Intercept(s) of a Line
Graph a Line
Slope-Intercept Form of a Line: y  mx  b
Slope of a Line m 
Point-Slope Form of a Line: y  y1  m  x  x1 
Standard Form of a Line: Ax  By  C
Parallel and Perpendicular Lines
Relations
Domain and Range of a Relation
Functions
Vertical Line Test
f -notation f  x  , g  x  , h  x  , etc.
Systems of Linear Equations
Solve a System of Linear Equations by:
1. Graphing
2. Elimination or Addition
3. Substitution
Recognize Consistent and Inconsistent Systems
Dependent and Independent Equations
Unit 2 Multiplying and Factoring Polynomials
Exam 2 – Sections 5.2, 5.3, 6.1, 6.2, 6.3, and 6.4
Polynomials
Monomial, Degree of a Monomial, Degree of a Polynomial
Adding, Subtracting, and Multiplying Polynomials
Like Terms
FOIL Method (used only to multiply two binomials)
Box or Table Method
Distributive Property
Factoring (to rewrite as a product)
Greatest Common Factor (GCF)
Factoring by Grouping
Unit 2 Multiplying and Factoring Polynomials (continued)
Factoring Trinomials of the Form ax 2  bx  c
Case 1: When a  1 example: x2  8x  12   x  6 x  2
Case 2: When a  1 example: 5x2 16 x  3   5x  1 x  3
Factoring Binomials
Difference of Two Squares: a2  b2   a  b  a  b 

Difference of Two Cubes: a 3  b3   a  b  a 2  ab  b 2

Sum of Two Cubes: a 3  b3   a  b  a 2  ab  b 2


Solving Polynomial Equations by Factoring
Zero-Factor Theorem
Unit 3 Rational Expressions and Rational Equations
Exam 3 – Sections 7.1, 7.2, 7.3, and 7.4
Simplifying Rational Expressions
Always look for common factors
Multiplying and Diving Rational Expressions
Cancel common factors before multiplying
Turn division into multiplication using the reciprocal of the divisor
P R P S
  
Q S Q R
Adding and Subtracting Rational Expressions
Least Common Denominator
Equivalent Expressions
Simplifying Complex Rational Expressions
Two Methods:
1. If we have a single rational expression divided by another single rational expression, we can
invert and multiply.
2. If a sum or difference of rational expressions occurs in the numerator and/or denominator, we
can multiply the numerator and denominator by the Least Common Multiple of all the
denominators.
Solving Equations Containing Rational Expressions
Use the Least Common Denominator to clear all fractions
Check for extraneous solutions
Unit 4 Radical Expressions and Radical Equations
Exam 4 – Sections 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, and 8.7
Evaluating Radicals
Radicand, Root Index, and Radical Sign
Square Roots, Cube Roots, Fourth Roots, etc.
a1 n  n a
am n  n am 
 
n
m
a
Rules of Exponents
The Product Rule, Quotient Rule, Power of a Product Rule, Power of a Power Rule, etc., are all valid
when the exponents are rational numbers.
Radical Arithmetic
Add, Subtract, Multiply, and Divide Radical Expressions
Simplify Radical Expressions
A Radical Expression is in simplest form if:
1. There are no factors in the radicand raised to powers greater than or equal to the root index.*
2. There are no fractions in the radicand.
3. There are no radicals in the denominator of the radical expression.
* If the root index is 2, this means there should be no perfect squares in the radicand
If the root index is 3, there should be no perfect cubes in the radicand
etc.
Rationalizing Denominators of Radical Expressions
Conjugates
Solving Radical Equations
Isolate the radical and then raise each side of the equation to the nth power (n = root index)
Check for extraneous solutions
Complex Numbers
The Imaginary Unit i
i  1 and i 2  1
Standard Form of a Complex Number
a  bi
Complex Number Arithmetic
Add, Subtract, Multiply, and Divide Complex Numbers
Complex Conjugates
Unit 5 Quadratic Equations
Exam 5 – Sections 9.1, 9.2, and part of 9.3
Quadratic Equations
Standard Form of a Quadratic Equation: ax 2  bx  c  0
Solve a Quadratic Equations using one of the following:
1. Factoring
2. Applying the Square Root Principle
 Quantity 
Note:
2
 number  Quantity   number
Quantity refers to some expression containing the unknown we want to solve for.
3. Completing the Square ( x 2  bx  c  0 )
i.
Divide the middle coefficient by 2
ii. Square the result from i
iii. Add the result from ii to both sides of the equation
iv.
Rewrite the perfect square trinomial as the square of a binomial and then apply the Square
Root Principle
4. The Quadratic Formula: x 
b  b2  4ac
2a