Rational Expressions

Definition and Properties

Rational expressions are fractions in which the numerator and denominator are polynomials. They are fundamental in algebra and precalculus, as they allow for the manipulation and simplification of polynomial relationships.

• Rational Expression: An expression of the form , where and are polynomials and .

• Domain: The set of all real numbers except those that make the denominator zero.

• Example: is defined for all .

Simplifying Rational Expressions

Simplification involves factoring polynomials and canceling common factors in the numerator and denominator.

• Factor both numerator and denominator.

• Cancel common factors.

• Order of operations: Follow PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

• Example: for .

Multiplying and Dividing Rational Expressions

To multiply or divide rational expressions, multiply/divide numerators and denominators, then simplify.

• Multiplication:

• Division:

• Example:

Radicals

Definition and Properties

Radicals involve roots, most commonly square roots. They are used to express numbers that are not perfect powers.

• Square root: is a number which, when squared, gives .

• Properties:

• Example:

Multiplying and Simplifying Radicals

Multiplication of radicals uses the property that the product of square roots equals the square root of the product.

• Example:

• Rationalizing denominators: Multiply numerator and denominator by a suitable radical to eliminate radicals from the denominator.

• Example:

Linear Equations

Definition and Solving Techniques

Linear equations are equations of the first degree, meaning the variable is raised only to the first power. They are foundational in algebra and precalculus.

• General form:

• Solving: Isolate the variable using inverse operations.

• Example:

Systems of Linear Equations

Systems involve two or more linear equations with multiple variables. Solutions are found where the equations intersect.

• Example: Solve

• Solution: Add equations: , then

Applications of Linear Equations

Linear equations model real-world relationships such as speed, distance, time, and financial calculations.

• Distance formula:

• Example: If a car travels at 60 mph for 2 hours, miles.

• Financial applications: Simple interest, loans, and investments can be modeled with linear equations.

Complex Numbers

Definition and Properties

Complex numbers extend the real numbers by including the imaginary unit , where . They are written in the form .

• Real part:

• Imaginary part:

• Example:

Operations with Complex Numbers

Complex numbers can be added, subtracted, multiplied, and divided using algebraic rules.

• Addition:

• Subtraction:

• Multiplication:

• Division:

• Complex conjugate: For , the conjugate is .

Practice Problems

• Add:

• Divide: Multiply numerator and denominator by the conjugate of the denominator:

Table: Properties of Complex Numbers

Summary

• Rational expressions, radicals, linear equations, and complex numbers are foundational topics in precalculus.

• Mastery of these concepts is essential for success in higher-level mathematics and real-world applications.