Ex - Add rational expressions with unlike denominators

TO ADD RATIONAL EXPRESSIONS, JUST LIKE ADDING FRACTIONS, WE MUST HAVE A COMMON DENOMINATOR. SO TO START THIS, I'M GOING TO GO AHEAD AND REWRITE THESE FRACTIONS AND PUT THE NUMERATORS AND DENOMINATORS IN PARENTHESES. SO WE'LL HAVE THE QUANTITY (X - 3) OVER THE QUANTITY (X + 6), AND WE'LL LEAVE QUITE A BIT OF SPACE HERE HORIZONTALLY, + THE QUANTITY (X + 7) ALL OVER THE QUANTITY X - 4. SO IN ORDER TO HAVE A COMMON DENOMINATOR, THE DENOMINATORS MUST CONTAIN THE SAME FACTORS. YOU'LL NOTICE HOW THIS FRACTION CONTAINS A FACTOR OF (X + 6), THIS FRACTION CONTAINS A FACTOR OF (X - 4) WHICH MEANS THE COMMON DENOMINATOR WOULD BE THE PRODUCT OF THESE TWO FACTORS. SO THIS FRACTION HERE MUST HAVE A FACTOR OF (X - 4) IN THE DENOMINATOR. AND WE CAN GO AHEAD AND MULTIPLY THE DENOMINATOR BY THE QUANTITY (X - 4) AS LONG AS WE DO THE SAME TO THE NUMERATOR. REMEMBER, ANYTHING OVER ITSELF IS EQUAL TO 1, SO THIS IS LIKE MULTIPLYING BY 1, PRODUCING AN EQUIVALENT FRACTION. THE SECOND FRACTION IS MISSING A FACTOR OF (X + 6), SO MULTIPLY BOTH THE DENOMINATOR AND THE NUMERATOR BY A FACTOR OF (X + 6). NOW, NOTICE THE DENOMINATORS CONTAIN THE SAME FACTORS, AND SO WE HAVE OUR COMMON DENOMINATOR. SO TO FIND THIS SUM, AGAIN, OUR DENOMINATOR IS GOING TO BE THE QUANTITY (X + 6) X THE QUANTITY (X - 4), AND OUR NUMERATOR IS GOING TO BE THE SUM OF THESE TWO PRODUCTS. SO WE'RE GOING TO LEAVE THE DENOMINATOR IN FACTORED FORM, AND THEN WE'RE GOING TO MULTIPLY OUT THE NUMERATORS SO THAT WE CAN THEN ADD THE LIKE TERMS. WHEN PERFORMING THIS MULTIPLICATION IN THE NUMERATOR, WE'LL HAVE FOUR PRODUCTS, ONE, TWO, THREE, FOUR. SO WE'RE GOING TO HAVE (X X X), THAT'S X SQUARED, (X X -4), THAT'S -4X, AND THEN (-3 X X), THAT'S -3X. SO WE'LL HAVE -7X OR - 7X AND THEN -3 X -4 IS +12, SO WE HAVE + 12 +, AND WE'LL FIND THIS NEXT PRODUCT. AGAIN, WE'LL HAVE FOUR PRODUCTS, ONE, TWO, THREE, AND FOUR. X X X IS X SQUARED. X X 6 IS 6X, AND 7 X X IS 7X, SO THAT'S GOING TO BE + 13X, AND THEN 7 X 6 IS 42, SO WE HAVE + 42. AND NOW BECAUSE WE'RE ADDING OUR NUMERATORS, WE JUST NEED TO COMBINE THE LIKE TERMS. AND WE HAVE 1X SQUARED PLUS 1X SQUARED, AND THESE TWO ARE LIKE TERMS, SO WE HAVE 2X SQUARED. THEN WE HAVE -7X + 13X. THAT'S GOING TO BE POSITIVE 6X. AND THEN WE HAVE +12 + POSITIVE 42. THAT'S GOING TO BE + 54. NOW WE DO WANT TO CHECK TO SEE IF THIS IS GOING TO SIMPLIFY. TO DO THIS, WE'LL HAVE TO FACTOR THE NUMERATOR. NOTICE HOW THE NUMERATOR DOES CONTAIN A COMMON FACTOR OF 2, SO WE HAVE 2 X THE QUANTITY, WE'D HAVE X SQUARED + 3X + 27 ALL OVER THE QUANTITY X + 6 X THE QUANTITY X - 4. NOW IT LOOKS LIKE THIS MAY FACTOR, BUT IT'S NOT GOING TO. THERE ARE NO FACTORS OF +27 THAT ADD TO +3, AND SINCE EVERYTHING IS IN FACTORED FORM, WE CAN SEE THIS RATIONAL EXPRESSION DOES NOT SIMPLIFY AND THEREFORE THIS WOULD BE OUR SUM. OKAY. I HOPE YOU FOUND THIS HELPFUL.