Algebra II

Advanced algebraic concepts including polynomials, rational expressions, and complex numbers.

Polynomials and Their PropertiesFactoring and Solving Polynomial EquationsRational Expressions and Equations
Complex Numbers and Imaginary UnitsPolynomial Functions and Their GraphsSolving Systems of Equations Algebraically
Modeling Real-World Data with PolynomialsEngineering and Physics ApplicationsFinance and Business Mathematics
Mastering Factoring TechniquesEfficient Equation Solving
Polynomials and Thei...Factoring and Solvin...Rational Expressions...Complex Numbers and ...Polynomial Functions...Solving Systems of E...Modeling Real-World ...Engineering and Phys...Finance and Business...Mastering Factoring ...Efficient Equation S...
Basic Concepts

Rational Expressions and Equations

Working with Rational Expressions

Rational expressions are fractions with polynomials in the numerator and denominator. You can simplify, multiply, divide, add, or subtract them—just like with numerical fractions!

Simplifying

Factor both the numerator and denominator and cancel any common factors. Always check for restrictions—values that would make the denominator zero.

Solving Rational Equations

To solve equations with rational expressions, find a common denominator, multiply both sides to eliminate denominators, then solve the resulting equation.

Why Rational Expressions Matter

Rational expressions are used in everything from calculating speed to understanding rates in chemistry.

Applications

Rational equations can describe things like mixing solutions, comparing speeds, or sharing resources.

Examples

  • Simplifying \( \frac{x^2 - 4}{x^2 - 2x} = \frac{(x + 2)(x - 2)}{x(x - 2)} = \frac{x + 2}{x} \) (as long as \( x \neq 0, 2 \))

  • Solving \( \frac{1}{x} + \frac{1}{2} = 1 \) gives \( x = 2 \ )

In a Nutshell

Rational expressions are fractions made from polynomials, and can be simplified or solved like regular fractions.

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Complex Numbers and Imaginary Units