Algebra Supporting Notes

When I started relearning pre-algebra, I used the pre-algebra section in Khan Academy as a guide. I went through the videos and found other written sources along the way. The problem with this approach is that I find Khan Academy's presentation lacking clarity. For this reason, I'm using this section to sort of reformat things by starting my conceptual understanding of the subject matter from scratch. 

I. What is Algebra?

Algebra is a branch of mathematics that deals with symbols (variables) and the rules for manipulating those symbols to represent and solve problems involving relationships between quantities. These symbols typically represent numbers, and they allow us to write mathematical expressions and equations in a generalized form. Key features include use of variables, expressions, equations, inequalities, etc. 

II. What are the different types of equations in beginning algebra?

1. Linear Equations

• Form: ax + b = 0

• Description: These equations involve variables with a power of 1 (linear) and are graphically represented as straight lines.

2. Quadratic Equations

• Form: ax² + bx + c = 0

• Description: These equations involve a variable squared (x^2) and are graphically represented as parabolas.

3. Absolute Value Equations

• Form: $\mathrm{\mid }ax+b\mathrm{\mid }= c$

• Description: These equations involve the absolute value of an expression, resulting in two potential solutions.

4. Rational Equations

• Form: $\frac{P(x)/}{Q(x) }= R(x)$
  

• Description: These equations involve fractions with polynomials in the numerator and/or denominator. Solutions must exclude values that make the denominator zero.

5. Radical Equations

• Form: √x + a = b

• Description: These equations include variables within a square root (or other roots). Solutions may need to be checked for extraneous results.

6. Exponential Equations

• Form: a² = b

• Description: These equations have variables as exponents. They often require logarithms to solve.

7. Inequalities (Optional in Some Courses)

• Form: $a\mathrm{x }+ \mathrm{b }> c$

• Description: These are similar to equations but involve inequality signs ($>, <, \ge , \le$).

8. Proportions

• Form: $\frac{\mathrm{a/}}{\mathrm{b }}= \frac{\mathrm{c/}}{d}$

• Description: These involve two ratios set equal to each other and are solved by cross-multiplication.

9. Systems of Equations

• Form:

  • Linear system:

    
    
    
    
    

• Description: These involve solving two or more equations simultaneously.

Understanding these types helps in identifying the approach and techniques needed to solve different algebraic problems.

III. Are linear expressions also polynomials?
Yes

A polynomial is any expression made up of variables and constants using only:

• Addition

• Subtraction

• Multiplication

• Non-negative integer exponents

A linear expression has the form:

ax+b

Where:

a and b are constants

x is the variable

The exponent on x is 1 (which is allowed in polynomials)

All linear expressions are polynomials,
but not all polynomials are linear expressions.