Math 30-1 Unit 2 Checklist

Unit 2: Rational Expressions and Equations Checklist

1. Rational Expressions

   • Define what a rational expression is: a ratio of two polynomials, $\frac{P(x)}{Q(x)}$, where $\text{Q(x)} \ne 0$.

   • Identify Non-Permissible Values (NPVs) of rational expressions:

     • Set the denominator equal to zero and solve for $x$.

     • Understand why NPVs exist (division by zero is undefined).

   • Simplify rational expressions:

     • Factor the numerator and denominator completely.

     • Cancel common factors.

     • State NPVs throughout the simplification process.

2. Operations with Rational Expressions

   • Multiplication

     • Factor all numerators and denominators.

     • Identify all NPVs from all original denominators.

     • Cancel common factors before multiplying.

     • Multiply remaining numerators and denominators.

   • Division

     • Factor all numerators and denominators.

     • Identify NPVs from the first denominator, the second denominator, and the second numerator (because it becomes a denominator when inverted).

     • "Multiply by the reciprocal" (flip the second fraction and change division to multiplication).

     • Proceed as with multiplication.

   • Addition and Subtraction

     • Find a Common Denominator (LCD):

       • Factor all denominators.

       • The LCD is the product of all unique factors raised to the highest power they appear in any single denominator.

     • Identify all NPVs from all factored denominators.

     • Rewrite each rational expression with the LCD.

     • Add or subtract the numerators (distribute negative signs for subtraction).

     • Keep the common denominator.

     • Simplify the resulting rational expression, if possible.

3. Rational Equations

   • Define a rational equation: an equation containing one or more rational expressions ($\frac{P(x)}{Q(x)} = \frac{R(x)}{S(x)}$ or sum/difference).

   • Identify all NPVs first (from all denominators in the original equation).

   • Solve rational equations:

     • Method 1: Eliminate denominators by multiplying both sides by the LCD.

       • Distribute the LCD to all terms.

       • Solve the resulting polynomial equation (linear, quadratic, etc.).

     • Method 2: If it's a proportion ($\frac{A}{B} = \frac{C}{D}$), cross-multiply ($A \cdot D = B \cdot C$).

       • Solve the resulting equation.

   • Check for extraneous roots: Any solution that is an NPV must be rejected. Valid solutions are those not on the NPV list.

4. Applications of Rational Equations

   • Solve word problems involving rational equations:

     • Set up equations based on given scenarios (e.g., work-rate problems, distance-speed-time, direct/inverse variation).

     • Work-rate formula: $\frac{1}{t<em>1} + \frac{1}{t</em>2} = \frac{1}{T_{total}}$.

     • Define variables clearly.

     • Formulate the rational equation.

     • Solve the equation, remembering to check for extraneous roots.

     • Interpret the solution in the context of the problem (e.g., units, logical answer).

5. Review and Practice

   • Ensure understanding of the difference between simplifying expressions and solving equations.

   • Practice factoring polynomials thoroughly.

   • Systematically identify NPVs at each step.

   • Perform operations with accuracy, especially with signs.

   • Verify solutions by substituting them back into the original equation, if time permits, especially for extraneous roots.