Overview
This lecture introduces rational algebraic expressions, explains how to evaluate them, and discusses how to determine values that make these expressions undefined.
Rational Algebraic Expressions
- A rational algebraic expression is in the form p/q, where p and q are polynomials.
- The denominator (q) must not be equal to zero, otherwise the expression is undefined.
- Examples: 4/(x-2), 4x/(x²+9), (x²+2x-35)/(3x+4).
Evaluating Rational Expressions
- To evaluate, substitute given values for the variable(s) and simplify.
- Example: For (x²-4)/(x-2) when x=0, the result is 2.
- Example: For (x²-4)/(x-2) when x=1, the result is 3.
- If the denominator after substitution does not become zero, the expression is valid.
Evaluating with Multiple Variables
- Substitute the given values for each variable and simplify.
- Example: For (a²-2b)/(a-b) when a=1, b=2, the result is 3.
Undefined Rational Expressions
- Expressions are undefined when the denominator is zero.
- To find such values, set the denominator equal to zero and solve for the variable(s).
- Example: For (x²-2)/(x-4), the expression is undefined when x=4.
- Example: For (x²-4)/(x-2), the expression is undefined when x=2.
- If the denominator has more than one variable, set it to zero and solve for the relationship.
- Example: For (x²-y)/(x-y), the expression is undefined when x = y.
Key Terms & Definitions
- Rational algebraic expression — An expression in the form p/q, where both p and q are polynomials and q ≠0.
- Undefined expression — An expression is undefined when its denominator evaluates to zero.
- Evaluate — To substitute values for variables and simplify the expression.
Action Items / Next Steps
- Practice substituting various values into rational expressions and identify values that make them undefined.