1. Algebra 2A 7.1 September 19, 2007

2. Objectives Simplify rational expressions Simplify rational expressions Identify rational functions. Identify rational functions. Simplify rational functions. Simplify rational functions. Graph rational functions. Graph rational functions.

3. A little vocabulary Expression: a combination of numbers and mathematical operations that does not contain an equal sign Expression: a combination of numbers and mathematical operations that does not contain an equal sign Function: an equation for which each input has one output Function: an equation for which each input has one output Rational: the ratio of two integers (for numbers), the ratio of two polynomials (for expressions or functions) Rational: the ratio of two integers (for numbers), the ratio of two polynomials (for expressions or functions) Simplify: non-reducible form Simplify: non-reducible form

4. What makes an expression undefined? x. x. x-5 x-5 When is a fraction undefined? What must the denominator equal? x-5=0x=5 This expression is undefined when x=5 because 5 is undefined. 0 DIVISION BY ZERO MAKES AN EXPRESSION UNDEFINED.

5. How do you evaluate a rational expression? 3x. 3x.2x-5 Evaluate the expression above for x=4. Plug 4 in for x. 3(4)2(4)-5 In the calculator, key in “3 * 4 / (2 * 4 -5)” and press enter. The answer is 4.

6. Simplifying rational expressions Fundamental principle of rational expressions P=PR Q=QRwhere Q≠0 and R≠0 So, when faced with a rational expression, look for a common factor in order to simplify. 1215 What is the common factor? 3 * 4 3 * 5 12 = 4 15 5

7. More simplifying 4x 2 y3x-6 12xy 2 x 2 -4 x+25x 2 -5 x+3x 2 -4x-5

8. Something to help you factor a-b = -(-a+b) = -(b-a) = -1(b-a) a-b=-(b-a) = -1 b-ab-a How does this work? 2x-4 4-x 2

9. Steps for simplifying any rational expression 1. Completely factor both the numerator and the denominator. 2. Divide the numerator and denominator by all common factors. 3. The resulting expression will be in the simplest form (in the lowest terms).

10. Rational functions f(x) = P Q Where P and Q are polynomials are Q ≠ 0 Which of the following are rational functions? f(x) = 3x 2 -2x+5 2x-5 2x-5 f(x) = 3x 3 + 3√x

11. Simplifying rational functions f(x)=x 2 +2x +1 x+1 x+1 What makes the denominator equal zero? This represents a hole in the graph. Now simplify: f(x) = (x+1) (x+1)=x+1 x+1 x+1 Plug the “hole” into the simplified expression to find its ordered pair -1 +1 =0 The ordered pair is (-1, 0)

12. Graphing a rational function f(x)=x 2 +2x +1 x+1 x+1 Simplified: x+1 Where is the hole? Mark the hole on the graph with an open circle.