CLEP Precalculus

CLEP Precalculus covers essential mathematical concepts and skills necessary for success in calculus and higher-level mathematics.

Functions and Their PropertiesPolynomials and Rational ExpressionsExponential and Logarithmic Functions
Trigonometric Functions and ApplicationsAnalytic GeometrySequences, Series, and Mathematical Induction
Modeling Population GrowthAnalyzing Sound WavesOptimizing Profits with Quadratic Functions
Mastering Calculator SkillsTime Management and Question PrioritizationActive Note-Taking and Formula Memorization
Functions and Their ...Polynomials and Rati...Exponential and Loga...Trigonometric Functi...Analytic GeometrySequences, Series, a...Modeling Population ...Analyzing Sound Wave...Optimizing Profits w...Mastering Calculator...Time Management and ...Active Note-Taking a...
Basic Concepts

Polynomials and Rational Expressions

Polynomials: The Building Blocks

Polynomials are expressions made by adding, subtracting, and multiplying variables and numbers. They show up everywhere in mathematics!

  • Standard Form: Arrange terms by descending exponents, such as \( 2x^3 - 5x^2 + 4x - 7 \).
  • Degree: The highest exponent, which tells you the general shape of the graph.

Operations with Polynomials

  • Addition/Subtraction: Combine like terms.
  • Multiplication: Use distributive property (FOIL for binomials).
  • Factoring: Reverse multiplication to break down expressions.

Rational Expressions

These are fractions where the numerator and/or denominator are polynomials, like \( \frac{x^2 - 1}{x + 1} \). Simplifying rational expressions often involves factoring and reducing.

Real-World Impact

Engineers use polynomials to design roller coasters, while rational expressions help in calculating rates, such as speed or efficiency.

Examples

  • Adding \( (2x^2 + 3x) + (x^2 - x) = 3x^2 + 2x \).

  • Simplifying \( \frac{x^2 - 4}{x + 2} = x - 2 \) for \( x eq -2 \).

In a Nutshell

Polynomials and rational expressions are algebraic tools used to model and solve a wide range of problems.

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Exponential and Logarithmic Functions