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Multiplying and dividing Rational functions with factoring first. There are 2 to doing these problems: Factoring And Simplifying numerators with denominators Lets start with simplifying a rational expression. (x+6) Factor the numerator: C ancel out the common terms in the numerator and denominator. Factor the denominator: (x-4) x 2 x 24 2 x 7x 6 2 (x+6) Original problem (x+1) The answer is… x4 x 1 To simplify a binomial they must be the exact same. Monomial variables subtract exponents. Numbers divide or reduce. Common term Difference of Squares Factor Grouping Trinomial Multiplying a rational expression. Factor all the numerators : 2x(x-5) (x+3) 2 x 10 x x 3 2 2 x 25 2x 2 (x+5)(x-5) Factor all the denominators: 2 2x Multiplying a rational expression. 2x(x-5) (x+3) 2 x 10 x x 3 2 2 x 25 2x 2 (x+5)(x-5) Cancel out or simplify the common terms in the numerator and denominator no matter where they are. 2 2x x The answer is… x3 x( x 5) *** very important an x by itself can not cancel with an x that is being added or subtracted to a number 2 4x 2 20 x 12 x 2 4x 4 x (5 x 3) x 5x 3 2 x 10 x 2 3x 16 x 5 2 2 x ( x 5) (3 x 1)( x 5) 2x 3x 1 x 2 3x 4 2 x 2 4 x 2 2 x 4x 4 x 4x 3 ( x 4)( x 1) 2 x( x 2) ( x 2)( x 2) ( x 3)( x 1) 2 x( x 4) ( x 2)( x 3) Practice x - 9x + 8 4x · 2 2 3x - 24x x -1 2 3 x 2 3x 10 x 2 10 x 21 2 x 2 x 15 Dividing Rational expressions You will flip over this one. Literally you flip the rational that comes AFTER the division sign then… The division becomes multiplication and you do the same thing as the last problems. 4x x 2x 2 5 x 20 x 6 x 8 2 4x x 6x 8 2 5 x 20 x 2x 2 4x ( x 4)( x 2) 5( x 4) x( x 2) 4 5 3x 13x 4 4 x 16 2 x 4 x2 2 (3x 1)( x 4) ( x 2) ( x 2)( x 2) 4( x 4) 3x 1 4( x 2) Practice x -13x +36 x + 3x - 4x -12 ¸ 3 4 2 2 x -14x - 32 x - 4x + 2x - 8 4 2 3 2 Adding and Subtracting Rational Functions You still get to… … but only the denominator Factor Basically you find the Least common denominator (which will include factors) and multiply the numerators by the missing factors. 2 + 5 2 5 2 2 3x 3x 2 3x LCD since both denominators are the same 7 2 3x This rational needs to multiply by x. This rational needs to multiply by (x-4) 1 x 1 x 2 2 3x 3x 12 x 3x2 3x(x-4) Multiply each numerator by what the denominator is missing. LCD: 3x2 (x-4) must contain all of the denominators but nothing extra And the solution is… 1( x 4) x( x) x 4 x 2 2 3x ( x 4) 3x ( x 4) 2 x x4 2 3x ( x 4) 2 How do I get the LCD? First I find the smallest number that all the coeficiants will divide into evenly. Ex. 4 and 10 would be 20 because that is the smallest number that 4 and 10 divide into evenly. Next I take any base variable that is any denominator and use the one with the largest exponent Ex. x, x2 , and x3 the LCD is x3 Lastly any binomial in the denominator I do the same as a variable. Ex (x + 2)(x + 2) and (x + 2)(x – 1) the LCD is (x + 2)2 (x – 1) because there are 2 (x + 2)’s in one denominator. 3 1 4x 7 lcd : 28 x 3(7) 1(4 x) 28 x 28 x 21 4 x 28 x x 1 6 2 2 x 4x 4 x 4 factor : ( x 2)( x 2) ( x 2)( x 2) lcd : ( x 2)( x 2)( x 2) ( x 1)( x 2) 6( x 2) 2 ( x 2) ( x 2) ( x 2) 2 ( x 2) x 2 x 2 6 x 12 ( x 2) 2 ( x 2) x 2 7 x 14 ( x 2) 2 ( x 2) Practice 1 1 + x y x -3 3 + 2 2 x - 2x - 3 x + 3x + 2