Simplifying Rational Expressions MATH 018 Combined Algebra S. Rook

2 Overview Section 7.1 in the textbook: –Evaluate a rational expression –Domain of rational expressions Find where a rational expression is undefined –Simplify rational expressions –Simplify equivalent expressions

Evaluating a Rational Expression

4 Rational Expression: an expression of the form P / Q where P and Q are polynomials and Q ≠ 0 To evaluate a rational expression: –Substitute the given values for the variable(s) in the rational expression –Simplify the final answer!

Evaluating a Rational Expression (Example) Ex 1: Evaluate for the given value of x and simplify: a) ; when x = 6 b) ; when x = -2 5

6 Domain of Rational Expressions

77 Domain: set of allowable values for x The domain can also mean those values where the rational expression is UNDEFINED A rational expression can be viewed as a fraction –When is a fraction undefined? Basically an exercise in factoring

Domain of Rational Expressions Ex 2: Find the values where the rational expression is undefined: a) b) 8

9 Simplifying Rational Expressions

10 Simplifying Rational Expressions Consider simplifying 20 / 30 using prime factorization Works the same way with rational expressions –i.e. completely factoring a polynomial is the equivalent of prime factorization –Factor the numerator and denominator –Divide out common factors For polynomials other than monomials, common factors MUST have the SAME terms AND the SAME signs

Simplifying Rational Expressions (Continued) When dealing with a polynomial OTHER than a monomial: –Either divide out EVERYTHING –Or divide out NOTHING AT ALL This is a common mistake!!! 11

Simplifying Rational Expressions (Example) Ex 3: Simplify: a) b) 12

Simplifying Equivalent Expressions

14 Simplifying Equivalent Expressions Always be on the lookout for EQUIVALENT FACTORS (same factors just written in a different form) in a rational expression –Cleans up the final answer –Makes the process of adding or subtracting rational expressions much easier Recall the commutative property of addition x + 7 and 7 + x are equivalent –Thus, what does (x + 7) / (7 + x) simplify to?

15 Simplifying Equivalent Expressions (Continued) Factoring out a negative SOMETIMES results in two equivalent factors: –What would (x – 5) / (5 – x) simplify to? –What about (x + 5) / (x – 5) ?

Simplifying Equivalent Expressions (Example) Ex 4: Simplify completely: a) b) c) 16

17 Summary After studying these slides, you should know how to do the following: –Evaluate a rational expression –Determine where a rational expression is undefined –Simplify a rational expression –Recognize and simplify equivalent forms of a rational expression Additional Practice –See the list of suggested problems for 7.1 Next lesson –Multiplying & Dividing Rational Expressions (Section 7.2)