This blog contains part one (of two) of a very brief review of concepts in Algebra that should familiar with the user hoping to learn Calculus. If any of these concepts are problematic I would advise the user to delve deeper into that subject matter until they are comfortable with exercises that are designed to test the user’s proficiency in those areas. I have provided links to websites that will go into greater detail and some exercises that the user should understand without any difficulty. If the user is having problems with the exercises, it would be advisable to explore the links provided.

This part will cover:

• Numbers, expressions, equations, inequalities and functions
• Factoring
• Graphing

Numbers, expressions, equations, inequalities and functions:

Some basic terminology…

• Natural numbers: The set of numbers {1,2,3,4…} containing all positive integers. The numbers that start with 1 and continue to add one to the previous number ad infinitum.
• Whole numbers: The set of numbers {0,1,2,3…} containing all natural numbers and 0. There is some disagreement to the exact definition.
• Integers: The set {… -4,-3,-2,-1,0,1,2,3,4…}. The set contains all positive natural numbers, their negative equivalents and including 0.
• Rational numbers: Numbers that can be written as a ratio (x/y), where x and y are integers and y != 0 (!= doesn’t equal).
• Irrational numbers: A number that is non-repeating and does not terminate (ex. pi = 3.14159…, √ 2 = 1.41421…).

• Algebraic expression: A combination of numbers and variables connected through mathematical operations (+ – * / ^ and roots)  (ex. 3x²+4).
• Term: Products and quotients of numbers and variables with sign included (ex. 2xy, -4x²y, x/2).

• Monomial: An algebraic expression with one term.
• Binomial: ” ” with two terms.
• Trinomial: ” ” with three terms.
• Multinomial: An algebraic expression with two or more terms.
• Polynomial: ” ” where variables do NOT appear in the denominator, under radical signs nor raised to non-positive or non-integer powers.
• Degree: For a monomial the degree is the sum of all the exponents of the variables (ex. 2x²y⁴ = 6^thdegree). A polynomial’s degree is the term with the highest degree (ex. 4x+y²+z = 2^nd degree, 3x²y⁴z + x⁶ – 2x + 1 is a 7^th degree polynomial).

• Rational expression: An expression that can be written as the ratio or quotient of two polynomials where the denominator does NOT equal 0 (ex. (x²-16)/(4y) ).
• Radical expression: An expression containing radicals (ex. ⁴√625, y + √x).

• Equation: Statement of equality between two algebraic expressions (ex. 4x = 2x – 3)
• Linear equation: An equation of the first degree. ax + b = 0 (a != 0)
• Quadratic equation: An equation of the second degree. ax² + bx + c = 0 (a != 0)
• Radical equation: An equation where the variable appears in a radicand.

• Inequality: A relation showing the relationship between two or more quantities using <,<=,>,>=.
• Polynomial inequality: An inequality where all terms in the unknown are polynomials.
• Rational inequality: An inequality where all terms in the unknown are rational expressions.

• Functions: A relation between variables so each values of one is paired with exactly one value for another.

Factoring:

Factoring is the process of finding the primes or irreducible polynomials given an expanded polynomial. I will keep this section simple by only considering polynomials with integer coefficients.

1. Factor out the greatest common factor of the polynomial. 6x² – 36x –> 6x is the greatest common factor —> factors to: 6x(x-6)

2. Special cases: Always keep an eye out for special cases as it will save you time. 

• Difference of two squares a²-b² = (a+b)(a-b)

3. Brute force trinomials (ax² + bx + c)

1. Begin by writing ( ?x + ?)(?x + ?) Remember the product of the coefficients of both x’s will equal “a” and the product of the constants will be “c”. 
2. Write out the possible combinations of products that will equal “a” and “c”.
3. Fill in (?x + ?)(?x + ?) with those combinations until you can exactly reproduce the original trinomial. Be mindful of negatives… remember your solution could be any of the following depending on the original trinomial  (?x + ?)(?x + ?), (?x + ?)(?x – ?) or (?x – ?)(?x – ?).

Example: Factor 3x² + 16x + 5

• All positive coefficients so the answer type will be of the form (?x+?)(?x+?)… there will be no negatives.
• Write out products for a : 1 and 3 and c: 1 and 5
• Test combinations  (3x+5)(x+1) = 3x²+8x+5  Nope. (3x+1)(x+5) Yes!

Graphing

Cartesian coordinate system: Know how to graphic basic functions and understand the four quadrants (i.e. I: x>0,y>0, II:x<0,y>0…)

Distance formula d=sqrt( (x₂-x₁)² + (y₂-y₁)²) ) Midpoint = (x₁+x₂)/2,(y₁+y₂)/2

Slope of a line m=rise/run=(y₂-y₁)/(x₂-x₁)

Horizontal lines m =0, vertical lines m=undefined, parallel lines have m₁=m₂ (slopes equal) Perpendicular lines have m₁m₂ = -1

Symmetry

• Y-axis: Substitution of -x for x yields same equation
• X-axis: Substitution of -y for y yields same equation
• Origin symmetric: Substitution of BOTH -x for x AND -y for y yields same equation

Equations of a line:

• Slope-point form y-y₀=m(x-x₀)
• Slope intercept form y=mx+b
• Two-point form y-y₁ = ((y₂-y₁)/(x₂-x₂))(x-x₁) provided x₂ != x₁

Graphing functions:

• Vertical shift: y=f(x)+c <— y=f(x) —> y=f(x) -c
• Horizontal shift: y=f(x+c) <— y=f(x) —> y=f(x-c)
• Vertical stretch: y=f(x) —> y=cf(x) where c >1
• Vertical compression: y=f(x) —> ycf(x) where 0<c<1
• Reflection y=f(x) —> y=-f(x)

Piecewise function is a function that is described by more than one algebraic expression…

f(x) = x if x<0; x+4×2 if 0<=x<4; x if 4<=x

Conic sections:

• Circle (x-h)² + (y-k)² = r²
• Ellipse (x-h)²/a² + (y-k)²/b² = 1
• Hyperbola x²/a² – y²/b² = 1 y²/a² – x²/b² = 1

Here are a few good links to websites that go into these topics with more detail:

http://en.wikipedia.org/wiki/Elementary_algebra

http://www.algebasics.com/

Next up… same sample exercise and Algebra Review Part II (Linear equations, polynomial and rational functions, and advanced topics).

Calculus Review is a blog designed to provide users with a step-by-step guide through the mathematical topic of calculus.

Calculus Review will be organized in a lesson-by-lesson format, where each individual post will cover one topic starting from the very basics. Each sequential lesson will build on the previous lessons but will also try and be as self-contained as possible. The benefits of this site being the ability to quickly connect current ideas to past topics. I envision lessons that not only teach and give examples to practice but also how the topic-at-hand applies to real-world situations. I hope a helpful and positive community develops alongside the posts to help and encourage one another.

As with all new subjects, there are some prerequisites required for Calculus. These previously learned skills and concepts will ensure the user has enough background information to make learning Calculus an enjoyable and smooth learning process. These skills don’t guarantee that learning calculus will be easy but it does give the user confidence that they have the tool-set to tackle the lessons at hand. Learning is not an easy process but with some time and effort it is one of the most rewarding experiences we can give ourselves.

The prerequisites for Calculus Review are:

• Algebra
• Analytic Geometry
• Elementary Functions
• Geometry
• Trigonometry

Don’t worry if you are unfamiliar or even a little rusty with these subjects. It is even possible you know them but under another name. For instance Analytic Geometry is also known as Coordinate or even Cartesain Geometry. It is just a fancy way of saying it is the study of geometry within an algebraic context (using algebraic representation and manipulation techniques). Before I start lessons in Calculus, I will begin with a quick review of these subjects. I will also provide links to good sources of more information on each of the subjects so you can fill in any gaps of knowledge you might have. This is what this blog is about… linking to other resources so the user can explore on their own with the basics or more advanced techniques.

My first real post in the subject matter will start with a review of Algebra.