Manipulating of Rational Expressions

Manipulating of Rational Expressions

Expressing Rational Expressions

  • A rational expression is one that can be expressed as the fraction of two polynomials.
  • Working with rational expressions is similar to working with regular fractions.

Simplifying Rational Expressions

  • Always check if top and bottom of the fraction can be factored. If a same factor is found in both, it can be canceled out.
  • Example: (3x^2 -12x)/3x simplifies as (3x(x - 4))/3x and further simplifying gives x - 4, after cancelling out 3x.

Adding and Subtracting Rational Expressions

  • To add or subtract rational expressions, they must have the same denominator, just like regular fractions.
  • If two rational expressions do not have the same denominator, find a common denominator and transform them.
  • Whenever you perform the operations, reflect the changes in the least common denominator.

Multiplying and Dividing Rational Expressions

  • When multiplying, simply multiply the numerators together and the denominators together. Aim to simplify afterwards.
  • When dividing, remember to “flip” the second expression (reciprocal) and then multiply.

Solving Equations with Rational Expressions

  • Similar to equations with fractions, to clear the fraction, multiply by the common denominator.
  • Be careful of values that make the denominator zero. Always cross-check with the original equation since these can be extraneous solutions.

Graphing Rational Functions

  • Sketching involves identifying key features like vertical asymptotes, horizontal asymptotes, and holes if any.
  • Vertical asymptotes are found where the denominator equals zero, and horizontal asymptotes depend on the degree of the polynomials.

The Golden Rule: Always simplify a rational expression but never cancel terms.

Remember to check your solutions in the original question to ensure they don’t make any denominator equals zero, as they would be undefined.