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  • - IN ORDER TO ADD OR SUBTRACT RATIONAL EXPRESSIONS,

  • WE MUST HAVE THE SAME, OR LIKE, DENOMINATORS.

  • IN THESE TWO EXAMPLES,

  • NOTICE THE DENOMINATORS LOOK ALMOST THE SAME

  • BUT NOT QUITE THE SAME.

  • NOTICE HERE WE HAVE A +X.

  • HERE WE HAVE A -X.

  • HERE WE HAVE A NEGATIVE, HERE WE HAVE A +2.

  • SO THESE DENOMINATORS ARE ACTUALLY OPPOSITES.

  • SO TO DEAL WITH OPPOSITE DENOMINATORS,

  • WE CAN FACTOR OUT A NEGATIVE, OR -1,

  • AND FROM ONE OF THE DENOMINATORS

  • TO MAKE THE DENOMINATORS LOOK ALMOST THE SAME.

  • SO WE'LL LEAVE THIS FIRST FRACTION THE SAME,

  • BUT WE ARE GOING TO INCLUDE THE NUMERATOR AND DENOMINATOR

  • IN PARENTHESES.

  • SO WE'LL HAVE THE QUANTITY (12X + 1)

  • ALL OVER THE QUANTITY (X - 2) PLUS--

  • HERE WE HAVE THE QUANTITY 7X + 3.

  • AND NOW HERE WE'RE GOING TO FACTOR OUT A NEGATIVE OR -1.

  • I'M GOING TO USE A NEGATIVE,

  • BUT IT'S JUST GOING TO CHANGE THE SIGN OF THESE TWO TERMS.

  • SO INSTEAD OF HAVING A -X THIS WILL BE A +X,

  • AND INSTEAD OF BEING A +2 THIS WILL BE -2.

  • NOTICE IF WE DISTRIBUTE THE NEGATIVE,

  • WE WOULD HAVE THE SAME EXPRESSION,

  • THOUGH THE ORDER WOULD BE DIFFERENT.

  • NOW LET'S TALK ABOUT NEGATIVE FRACTIONS FOR A MOMENT.

  • IF I HAD, FOR EXAMPLE, -2/3,

  • THIS NEGATIVE SIGN CAN BE OUT IN FRONT OF THE FRACTION,

  • IT CAN BE IN THE NUMERATOR, OR IT CAN BE IN THE DENOMINATOR.

  • ALL OF THESE ARE EQUIVALENT,

  • SO WHAT WE'LL DO NOW IS JUST MOVE THIS NEGATIVE SIGN

  • UP INTO THE NUMERATOR,

  • AND WHEN WE DO THIS, NOTICE HOW THEN--

  • THE DENOMINATORS WILL BE THE SAME.

  • SO WE'LL HAVE THE QUANTITY (12X + 1)

  • ALL OVER THE QUANTITY (X - 2) PLUS--

  • OUR DENOMINATOR IS NOW GOING TO BE THE QUANTITY (X - 2),

  • AND THE NUMERATOR IS NOW GOING TO BE -(7X + 3).

  • NOW THAT WE HAVE LIKE DENOMINATORS,

  • THE DENOMINATOR IS GOING TO STAY THE SAME,

  • AND THEN WE'LL ADD THE NUMERATORS.

  • WE ARE GOING TO CLEAR THE PARENTHESES,

  • SO NOW WE CAN THINK OF DISTRIBUTING A +1 HERE

  • AND DISTRIBUTING A -1 HERE.

  • SO WE'D HAVE 12X + 1.

  • THIS IS GOING TO BE + A -7X OR -7X.

  • THIS WILL BE + A -3 OR - 3.

  • NOW, WE'LL COMBINE THE LIKE TERMS IN THE NUMERATOR.

  • WE STILL HAVE THE QUANTITY (X - 2) IN THE DENOMINATOR.

  • NOTICE OUR NUMERATOR IS GOING TO BE 12X - 7X.

  • THAT'LL BE 5X

  • AND 1 - 3 IS EQUAL TO -2,

  • SO WE HAVE 5X - 2 IN OUR NUMERATOR.

  • NOW, WE DO WANT TO TRY TO SIMPLIFY THIS FRACTION,

  • BUT SINCE A NUMERATOR DOES NOT FACTOR, THIS DOES NOT SIMPLIFY.

  • WE CANNOT SIMPLIFY THESE -2'S, FOR EXAMPLE,

  • BECAUSE WE CANNOT SIMPLIFY ACROSS ADDITION OR SUBTRACTION.

  • BUT SINCE THE NUMERATOR AND DENOMINATOR

  • DO ONLY CONTAIN ONE FACTOR,

  • THE PARENTHESES ARE OPTIONAL.

  • WE COULD WRITE THIS AS 5X - 2 ALL OVER X - 2.

  • EITHER OF THESE TWO FORMS ARE ACCEPTABLE.

  • NOW LET'S TAKE A LOOK AT OUR SECOND EXAMPLE

  • WHEN WE HAVE SUBTRACTION.

  • AGAIN, NOTICE HOW THE DENOMINATORS ARE OPPOSITES.

  • SO FOR THE FIRST STEP,

  • WE'RE GOING TO INCLUDE PARENTHESES EVERYWHERE.

  • SO WE'LL HAVE THE QUANTITY (X + 1)

  • OVER THE QUANTITY (X - 5)

  • MINUS THE QUANTITY (2X + 14) OVER--

  • AND AGAIN, BECAUSE OUR DENOMINATORS ARE OPPOSITES,

  • HERE WE'RE GOING TO FACTOR OUT A NEGATIVE,

  • AND THIS WILL BE A +X AND A -5.

  • NOW REMEMBER, THIS JUST MEANS THIS FRACTION IS NEGATIVE.

  • AND REMEMBER, SUBTRACTING A NEGATIVE

  • IS THE SAME AS ADDING A POSITIVE.

  • SO NOW WE CAN REWRITE THIS AS AN ADDITION PROBLEM.

  • WE WOULD HAVE THE QUANTITY (X + 1)

  • ALL OVER THE QUANTITY (X - 5) PLUS THE QUANTITY (2X + 14)

  • ALL OVER THE QUANTITY (X - 5).

  • AND NOW THAT WE HAVE A COMMON DENOMINATOR,

  • WE CAN ADD THE NUMERATORS.

  • AND AT THE SAME TIME WE'LL CLEAR THE PARENTHESES

  • WHICH WE CAN THINK OF JUST DISTRIBUTING A +1,

  • BUT IT'S NOT GOING TO CHANGE ANY SIGNS.

  • SO HERE WE'D HAVE (X + 1) + (2X +14).

  • NOW, WE'LL COMBINE LIKE TERMS.

  • 1X + 2X IS 3X.

  • 1 + 14 IS 15, SO + 15.

  • NOTICE HOW THE NUMERATOR DOES FACTOR.

  • THERE'S A COMMON FACTOR OF 3 HERE,

  • SO WE'D HAVE 3 TIMES THE QUANTITY (X + 5)

  • ALL OVER THE QUANTITY (X - 5).

  • NOTICE HOW THESE FACTORS ARE NOT THE SAME.

  • ONE IS A SUM, AND ONE IS A DIFFERENCE.

  • SO THIS DOES NOT SIMPLIFY.

  • SO WE'D LEAVE OUR ANSWER IN THIS FORM.

  • OKAY, I HOPE YOU FOUND THIS HELPFUL.

  •  

- IN ORDER TO ADD OR SUBTRACT RATIONAL EXPRESSIONS,

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B2 中高級

例:有理表達式的加減法------反分母 (Ex: Add and Subtract Rational Expressions - Opposite Denominators)

  • 63 8
    Hhart Budha 發佈於 2021 年 01 月 14 日
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