Algebra 2 - Unit 7: Rational Expressions

Simplifying Rational Expressions

• A rational expression comprises a numerator and a denominator, similar to rational numbers, but may include polynomials.
• Two rational expressions are equivalent if their cross-products are equal, or if they can be reduced to the same lowest terms.
• To reduce to lowest terms: factor the numerator and denominator completely, then cancel out common factors.

Multiplying and Dividing Rational Expressions

• Multiplication of rational expressions follows the same rules as fraction multiplication: multiply the numerators and denominators after factoring and simplifying.
• It is beneficial to cancel common factors before multiplying to simplify the process.
• For division, invert the second expression and multiply, ensuring all fractions are fully factored and simplified.

Adding and Subtracting Rational Expressions

• Like rational numbers, rational expressions can be added, subtracted, multiplied, and divided (but not by zero).
• To add rational expressions with identical denominators, add or subtract the numerators directly.
• For different denominators, convert to equivalent fractions using a common denominator.
• Identifying a common denominator involves factoring all denominators to see shared components.
• Subtraction requires careful distribution of the negative sign across the numerator.

Simplifying Complex Fractions

• Complex rational expressions can be simplified using two main methods: rewriting the numerator and denominator as single expressions and dividing, or multiplying by the least common multiple of the inner fractions' denominators.

Discontinuities of Rational Functions

• The domain of a rational function includes all values that do not make the denominator equal to zero.
• Discontinuities occur where input values are excluded from the domain, leading to vertical asymptotes or holes on the graph.
• A hole represents a removable discontinuity, often fixed by simplifying the rational expression.

Asymptotes of Rational Function

• The x-intercepts correspond to the zeros of the numerator after removing discontinuities.
• The y-intercept is found by substituting x = 0 into the function.
• Vertical asymptotes align with the zeros of the denominator after removing discontinuities.
• End behavior depends on the degrees of the numerator and denominator:
  • If the denominator's degree is higher, the function approaches zero, indicating a horizontal asymptote at y = 0.
  • If degrees are equal, the function approaches the ratio of leading coefficients.
  • If the numerator's degree exceeds the denominator's degree by one, a slant (oblique) asymptote occurs.
  • Greater differences in degrees lead to parabolic or higher-degree behavior.

Solving Rational Functions

• Rational equations can be solved using cross-multiplication to transform them into polynomial equations.
• Factoring before cross-multiplying aids in spotting common factors effectively.
• Extraneous solutions may arise during solving; these must be checked against the original problem to ensure validity.

Applications of Rational Equations

• Real-world scenarios such as mixtures, work rates, and comparisons can be represented with rational functions.
• Solving these rational equations typically yields solutions relevant to the modeled situation.
• When applying algebraic solutions to real-world contexts, it’s essential to interpret the meaning of the results accurately.

Test your knowledge of rational expressions with this flashcard quiz from Algebra 2, Unit 7. Learn how to simplify rational expressions and explore the similarities and differences between rational expressions and rational numbers. Perfect for mastering essential algebra concepts.