2. Warm-up Given these solutions below: write the equation of the polynomial: 1. {-1, 2, ½)

3. Rational Equations Section 2-6

4. Objectives I can simplify rational expressions I can find Domain Restrictions I can solve rational equations with one variable

5. Simplifying Rational Expressions Try and reduce numerator over denominator You will factor all numerators and denominators, then Reduce or cancel like terms

6. Domain of Rational Functions The domain of any rational function is all real numbers except where the following happens: –No x-value that makes denominator zero –No x-value that would be a discontinuity (hole)

7. EXAMPLE 1 Simplify a rational expression x 2 – 2x – 15 x 2 – 9 Simplify : x 2 – 2x – 15 x 2 – 9 (x +3)(x –5) (x +3)(x –3) = Factor numerator and denominator. (x +3)(x –5) (x +3)(x –3) = Divide out common factor. Simplified form SOLUTION x – 5 x – 3 = ANSWER x – 5 x – 3

8. GUIDED PRACTICE for Examples 1 and 2 x 2 – 2x – 3 x 2 – x – 6 5. (x – 3)(x + 1) (x – 3)(x + 2) x 2 – 2x – 3 x 2 – x – 6 = Factor numerator and denominator. Divide out common factor. x + 1 x + 2 = Simplified form = (x – 3)(x + 1) (x – 3)(x + 2) SOLUTION x + 1 x + 2 ANSWER

9. GUIDED PRACTICE for Examples 1 and 2 2x 2 + 10x 3x 2 + 16x + 5 6. 2x 2 + 10x 3x 2 + 16x + 5 (3x + 1)(x + 5) 2x(x + 5) = Factor numerator and denominator. Divide out common factor. 2x2x 3x + 1 = Simplified form (3x + 1)(x + 5) 2x(x + 5) = ANSWER 2x2x 3x + 1 SOLUTION

10. Adding & Subtracting Rational Expressions MUST have a COMMON DENOMINATOR You will factor all denominators, then find the Common Denominator Reduce or cancel like terms

11. Basic Fraction 6 6 x2 4 x3 3 4 + 3 6

12. (x+2) x(x+2) (x+3) 2(x+3) x(x+2) - 2(x+3)

13. 2 2 2(x-5) 1 1 x - 7 2(x-5) - 1(x-7)

14. Solving Rational Equations Two basic methods 1. Set equation equal to ZERO and then get Common Denominator 2. Two ratios equal means you can Cross Multiply to solve them

15. Cross Multiplication Method

16. Cross Multiplication Ex2

17. Set Equation to ZERO 2 2 2(x+1) (x-2) 5x(x-2) 6 6 6 2(x+1) – 5x(x-2) - 6 6(x-2) Next Slide

18. Problem Continued MUST CHECK ANSWERS x = 2 does not work

19. Extraneous Solutions Extraneous solutions are those that do not work when you plug them back into the original equation. Usually they don’t work because they make the Denominator zero

20. Homework Worksheet 5-1