Transcript x + 2
October 31st copyright2009merrydavidson Simplifying Rational Expressions What is the difference between a factor and a term? An expression has NO equal sign. ( x 5)(2 x 3) (3x 5)( x 2) TERMS are separated by +, - signs. FACTORS are separated by multiplication sign. The REDUCTION PROPERTY OF FRACTIONS You are allowed to divide out (cancel) common factors (not common terms) that appear in both the numerator and the denominator. You may NOT cancel 4x 5 the “4x”’s. They are terms not factors! 4x 3 Example 2: 2 2x p 4 pr There are NO add or subtract Can you cancel signs so these anything out? are all factors. Why or Why not??? Reduce the 2/4 to ½ and cancel out the p’s. 2 2 x List domain x ; r0 restrictions. 2r 2r domain restrictions are called critical numbers. CRITICAL NUMBERS: • Found in the numerator by solving for x. • Found in the denominator by solving for x. 2x 3 (4 x 3)( x 5) • C.N. = 3/2 • C.N. = ¾ ,-5 Example 3: ( x 5)(2 x 3) (4 x 3)( x 5) DO domain restrictions and critical numbers before canceling. 3 List domain restrictions. 4 List critical numbers. 3 2 SIMPLIFY: 2x 3 4x 3 2x 3 3 ; x 4x 3 4 Example 4: x 3x 10 2 x 4 2 Can you cancel anything out? Why or Why not? These are all terms so no. Factor to see if anything will match to cancel. ( x 5)( x 2) ( x 2)( x 2) x5 x2 SIMPLIFY: List domain restrictions. -5,2,-2 List critical numbers. -5,2,-2 x5 ; x -2 x2 Example 5: For this example only simplify and list answer with domain restrictions. x3 x7 x 9 ( x 3)( x 7) 2 ( x 3)( x 3) ( x 3)( x 7) Remember to do domain restrictions BEFORE canceling!!! x3 ; x 3, -7 x7 Example 6: ( x 5)(2 x 3) (4 x 3)(5 x) (x – 5) and (5 – x) are opposites….. In order to cancel you must factor out a (-1) from one of those terms. 1( x 5)(2 x 3) (4 x 3)(5 x) 1(5 x)(2 x 3) (4 x 3)(5 x) 1(2 x 3) 2 x 3 3 ; x ,5 4x 3 4x 3 4 7. Solve the following rational equation. Step 1: Find the LCD. 3 6x LCD = 6 Step 2: Multiply each term by the LCD over 1. Step 3: Cancel 2 26 6 7 1 2 31 1 3x + 4 = 42 x = 38/3 Step 4: Re-write the equation. Step 5: Solve the equation. 8. Solve the following rational equation. LCD is: 6 2 3 6 x 1 26 6 x 1 2 31 1 3(x + 1) + 4 = 6x 3x + 3 + 4 = 6x 7 = 3x 7/3 = x Extraneous Solutions: You are not allowed to have a zero in the denominator of a fraction. Therefore, If you get x = 5 and 5 would make the denominator = 0, 5 would be an extraneous solution. In other words, if algebraically you get a solution, but that makes the denominator zero it is called an extraneous solution. For any fraction with a variable in the denominator you must list domain restrictions first (that is what x can not be). 9. Solve the following rational equation. There is a variable in 3 8 13 5x 5x 5x the denominator. 1 x 51 x 1 What can x not be? x =0 LCD is 5x By using the LCD 15 + 8x = 65 and canceling you eliminate the 8x = 50 denominator. x = 25/4 Since the answer is not a domain restriction we are done. 10. Solve the following rational List domain equation. restrictions 5 2 x 5 ( x 5) ( x 5) ( x 5) FIRST. x 5 1 1 Solve Check for extraneous solutions. x 51 x5 1 x – 5 + 5 = 2x - 5 x = 2x - 5 5=x But x cannot be 5. the answer is: no real solution and 5 is extraneous. 11. Solve the following rational equation. x 1 x4 3 2 2 2 x 2 x x x x 3x 2 Factor everything possible. x 1 x4 3 x( x 2) x( x 1) ( x 2)( x 1) domain restrictions: x = 0, -2, -1 domain restrictions: x = 0, -2, -1 LCD: x( x 2)( x 1) 1 x 1 x4 3 x( x 2)( x 1) 1 x( x 2) x( x 1) ( x 2)( x 1) x( x 2)( x 1) 1 (x + 1)(x + 1) – (x + 2)(x + 4) = -3x x2 + 2x + 1 – x2 – 6x - 8 = -3x x = -7 12. Solve the following rational equation. x 3, 1, 4 3x 1 3 2x 5 2 2 2 x 4 x 3 x x 12 x 3x 4 ( x 3)( x 1)( x 4) 13 x 1 ( x 3)( x 1) ( x 3)( x 1)( x 4) 13 ( x 4)( x 3) ( x 3)( x 1)( x 4) 2 x 51 ( x 4)( x 1) Be sure that everything in the denominator gets canceled out 12. x 3, 1, 4 (x – 4)(3x – 1) – 3(x + 1) = (x + 3)(2x – 5) ( x 3)( x 1)( x 4) 13 x 1 ( x 3)( x 1) ( x 3)( x 1)( x 4) 13 ( x 4)( x 3) ( x 3)( x 1)( x 4) 2 x 51 ( x 4)( x 1) 3x2 – 13x + 4 – 3x – 3 = 2x2 + x - 15 x2 – 17x + 16 = 0 x = 1, 16 HW: WS 4-4