Ex - Subtract rational expressions with unlike denominators

- IN ORDER TO ADD OR SUBTRACT RATIONAL EXPRESSIONS JUST LIKE FRACTIONS, WE MUST HAVE A COMMON DENOMINATOR. SO, WHEN ADDING OR SUBTRACTING RATIONAL EXPRESSIONS, THE FIRST STEP IS TO MAKE SURE THE DENOMINATORS ARE IN FACTORED FORM. SO, THE FIRST STEP IS TO FACTOR THE DENOMINATOR OF THIS FRACTION HERE. AND WE'RE ALSO GOING TO PUT PARENTHESIS AROUND THE NUMERATORS AND DENOMINATORS. SO, FOR OUR FIRST FRACTION, WE'LL HAVE THE QUANTITY 2X PLUS 5 OVER-- NOW WE'LL GO AHEAD AND FACTOR OUR DENOMINATOR. WE'RE GOING TO HAVE TWO BINOMIAL FACTORS. THE FIRST TERMS WILL BE X AND X AND THEN WE WANT THE FACTORS OF NEGATIVE 18 THAT ADD TO POSITIVE 3, THAT'S GOING TO BE POSITIVE 6 AND NEGATIVE 3. SO, ONE FACTOR IS X PLUS 6 AND ONE FACTOR IS X MINUS AND THEN WE HAVE MINUS 3 ALL OVER A FACTOR OF X PLUS 6. AND NOW IN ORDER TO FIND THE COMMON DENOMINATOR, WE JUST NEED TO MAKE SURE THAT THE DENOMINATORS CONTAIN THE SAME FACTORS. WELL, NOTICE HOW THIS DENOMINATOR HERE CONTAINS A FACTOR OF X MINUS 3 AND THIS DENOMINATOR DOESN'T. SO, IF WE'RE GOING TO HAVE LIKE DENOMINATORS, THIS DENOMINATOR NEEDS A FACTOR OF X MINUS 3. AND AGAIN, MULTIPLY THE DENOMINATOR BY X MINUS 3, 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, BUT NOW WE HAVE A COMMON DENOMINATOR. SO, NOW THAT WE HAVE A COMMON DENOMINATOR, WE'LL WRITE THIS AS A SINGLE FRACTION WHERE OUR DENOMINATOR IS THE QUANTITY X PLUS 6 TIMES THE QUANTITY X MINUS 3 AND THEN WE'LL COMBINE OUR NUMERATORS. SO, WE HAVE THE QUANTITY 2X PLUS 5 MINUS-- LET'S GO AHEAD AND MULTIPLY THESE. SO, WE'D HAVE MINUS 3X MINUS REMEMBER HERE, WE JUST DISTRIBUTED THE 3. NOW, WE WANT TO CLEAR THESE PARENTHESES SO THAT WE CAN COMBINE THE LIKE TERMS AND BECAUSE WE HAVE SUBTRACTION HERE, WE DO HAVE TO BE CAREFUL. TO CLEAR THE PARENTHESIS FOR 2X PLUS 5, WE CAN JUST DROP THE PARENTHESIS OR THINK OF DISTRIBUTING A POSITIVE BUT SINCE WE'RE SUBTRACTING THE QUANTITY 3X MINUS 9, WE CAN THINK OF DISTRIBUTING A NEGATIVE SO, THE DENOMINATOR IS GOING TO STAY THE SAME. WE HAVE THE QUANTITY X PLUS 6 TIMES THE QUANTITY X MINUS AND AGAIN, HERE WE CAN JUST DROP THE PARENTHESIS OR DISTRIBUTE 1. SO, THAT'S JUST GOING TO BE 2X PLUS 5. BUT HERE, WE'RE DISTRIBUTING A NEGATIVE 1 BECAUSE OF THE SUBTRACTION, SO IT'S GOING TO BE MINUS 3X AND THEN IT'S NEGATIVE 1 TIMES NEGATIVE 9, SO IT'S GOING TO BE PLUS 9. AND NOW, WE'LL COMBINE THE LIKE TERMS IN THE NUMERATOR. HERE, WE HAVE 2X MINUS 3X AND HERE WE HAVE 5 PLUS 9. SO, AGAIN OUR DENOMINATOR STAYS THE SAME. WELL 2X MINUS 3X IS GOING TO BE NEGATIVE 1X OR NEGATIVE X AND THEN, WE HAVE 5 PLUS 9, THAT'S PLUS 14. THIS WILL NOT SIMPLIFY BECAUSE NEGATIVE X PLUS 14 DOES NOT FACTOR AND THEREFORE, THERE ARE NO COMMON FACTORS OTHER THAN 1 BETWEEN THE NUMERATOR AND THE DENOMINATOR. BUT SOMETIMES YOU WILL SEE THIS EXPRESSED IN A SLIGHTLY DIFFERENT WAY. IF WE WANTED TO, WE COULD FACTOR OUT A NEGATIVE OR NEGATIVE 1 FROM THE NUMERATOR. IF WE FACTOR OUT A NEGATIVE, IT'S JUST GONNA CHANGE THE SIGN OF THE X AND CHANGE THE SIGN OF THE PLUS 14. SO, WE'LL HAVE MINUS 14. SO, IT COULD BE EXPRESSED THIS WAY. OF COURSE, THESE ARE EQUIVALENT, SO IT SHOULDN'T MATTER, BUT THERE'S ALSO ONE OTHER WAY YOU MIGHT SEE THIS EXPRESSED. SOMETIMES WHEN WE HAVE A NEGATIVE FRACTION LIKE WE HAVE HERE, YOU'LL SEE THE NEGATIVE SIGN WRITTEN OUT IN FRONT OF THE FRACTION AND THEN WE'LL JUST HAVE X MINUS 14 IN THE NUMERATOR AND OUR FACTORS OF X PLUS 6 AND X MINUS 3 IN THE DENOMINATOR. SO, ALL THESE FRACTIONS ARE WRITTEN CORRECTLY, BUT I DO THINK IT'S IMPORTANT THAT WE NOTICE THAT THEY ARE EQUIVALENT BECAUSE DEPENDING ON YOUR INSTRUCTOR OR TEXT, YOU MAY SEE IT WRITTEN IN ANY OF THESE THREE WAYS. OKAY. I HOPE YOU FOUND THIS HELPFUL.