Precalculus Study Notes on Exponents and Inequalities

Study Notes

Negative Exponent Rule: ( a^{-n} = \frac{1}{a^n} )
Basic Steps:
1. Convert negative exponents to fractions.
2. Simplify fractions if necessary.
3. Combine like terms and express in standard form.
Example: ( x^{-2} = \frac{1}{x^2} )

Definition: An inequality involving the absolute value of a variable.
Types:
1. Linear Inequalities: ( |x| < a ) or ( |x| > a ).
2. Solutions:
  - For ( |x| < a ): ( -a < x < a )
  - For ( |x| > a ): ( x < -a ) or ( x > a )
Graphical Interpretation:
- Represents distances on a number line.
- Solutions can be visualized as segments or rays.

General Form: ( ax^3 + bx^2 + cx + d )
Methods:
1. Factoring by Grouping: Group terms and factor common factors.
2. Synthetic Division: Use to find roots and factor.
3. Rational Root Theorem: Test possible rational roots.
Example:
- For ( x^3 - 3x^2 - 4x + 12 ):
  - Factor to ( (x - 2)(x^2 - x - 6) )
  - Further factor ( (x - 2)(x - 3)(x + 2) )

Definition: A fraction where the numerator, denominator, or both are also fractions.
Steps to Simplify:
1. Find a common denominator for all fractions involved.
2. Combine fractions into a single fraction for both the numerator and denominator.
3. Simplify the resulting fraction if possible.
Example:
- Simplifying ( \frac{\frac{1}{2} + \frac{1}{3}}{\frac{1}{4}} ):
  - Combine numerator: (\frac{3}{6} + \frac{2}{6} = \frac{5}{6})
  - Rewrite: ( \frac{\frac{5}{6}}{\frac{1}{4}} = \frac{5}{6} \times 4 = \frac{20}{6} = \frac{10}{3} )

Negative exponent indicates the reciprocal, following the rule ( a^{-n} = \frac{1}{a^n} ).
Converting negative exponents to fractions simplifies expressions.
Example: The expression ( x^{-2} ) can be rewritten as ( \frac{1}{x^2} ).

Absolute value inequalities express the distance from zero on a number line.
Types include:
- Linear inequalities of the form ( |x| < a ) or ( |x| > a ).
Solution for ( |x| < a ) yields ( -a < x < a ).
Solution for ( |x| > a ) results in two conditions: ( x < -a ) or ( x > a ).
Graphically represented as segments (for ( < )) or rays (for ( > )) on a number line.

Cubic polynomials typically take the form ( ax^3 + bx^2 + cx + d ).
Common methods of factoring include:
- Factoring by grouping to extract common factors from grouped terms.
- Using synthetic division to find roots which helps in factoring the polynomial.
- The Rational Root Theorem assists in testing potential rational roots.
Example: The cubic ( x^3 - 3x^2 - 4x + 12 ) factors into ( (x - 2)(x^2 - x - 6) ), which can be further factored into ( (x - 2)(x - 3)(x + 2) ).

A complex fraction is one where the numerator, denominator, or both are fractions themselves.
Steps to simplify:
- Identify a common denominator for all constituent fractions.
- Combine fractions in both the numerator and denominator into a single fraction.
- Simplify the resulting fraction if possible.
Example: To simplify ( \frac{\frac{1}{2} + \frac{1}{3}}{\frac{1}{4}} ), combine terms in the numerator to obtain ( \frac{5}{6} ), then compute ( \frac{\frac{5}{6}}{\frac{1}{4}} = \frac{5}{6} \times 4 = \frac{20}{6} = \frac{10}{3} ).