- WELCOME TO A VIDEO ON MULTIPLYING AND DIVIDING
RATIONAL EXPRESSIONS.
WHAT YOU'LL FIND IS THAT MULTIPLYING RATIONAL EXPRESSIONS
IS JUST LIKE MULTIPLYING FRACTIONS.
SO, THE BASIC RULE IS TO FIND THE PRODUCT
OF TWO RATIONAL EXPRESSIONS,
YOU MULTIPLY THE NUMERATORS TOGETHER
AND THEN YOU MULTIPLY THE DENOMINATORS TOGETHER.
HOWEVER, BY NOW, WE KNOW
OUR PRODUCT MUST BE IN SIMPLIFIED FORM.
SO, IN ORDER TO RECOGNIZE THE COMMON FACTORS IN THE PRODUCT,
WE'LL WRITE THE NUMERATORS AND DENOMINATORS IN FACTORED FORM
TO SIMPLIFY BEFORE WE MULTIPLY.
LET'S GO AHEAD AND TRY THIS USING TWO FRACTIONS.
WE'RE GONNA WRITE EACH OF THESE IN PRIME FACTORED FORM,
7 IS PRIME, 15 WOULD BE 3 TIMES 5,
20 IS 4 TIMES 5 AND 4 IS 2 TIMES 2.
SO, WE HAVE 2 TIMES 2 TIMES 5, 21 IS 3 TIMES 7.
SO, WE KNOW FROM THE VIDEO
ON SIMPLIFYING RATIONAL EXPRESSIONS,
IF WE HAVE A COMMON FACTOR BETWEEN ANY NUMERATOR
AND ANY DENOMINATOR, IT SIMPLIFIES TO ONE
SO, HERE WE HAVE A 7 OVER 7, THAT WOULD EQUAL 1
AND THEN A 5 OVER A 5, THAT WOULD EQUAL ONE,
THERE ARE NO ADDITIONAL COMMON FACTORS,
SO NOW THE PRODUCT WILL BE IN SIMPLIFIED FORM.
SO, OUR NUMERATOR WILL BE 2 TIMES 2, THAT'S 4
AND OUR DENOMINATOR WILL BE 3 TIMES 3, WHICH IS EQUAL TO 9.
SO, I'LL FOLLOW THE SAME PROCEDURE ON THIS NEXT PROBLEM.
EVEN THOUGH THERE ARE SOME SHORTCUT METHODS
FOR SIMPLIFYING HERE BEFORE WE MULTIPLY,
I'M GOING TO GO AHEAD AND EXPAND ALL OF THESE.
SO, 5X CUBE Y WOULD BE 5 TIMES 3 FACTORS OF X TIMES Y,
4Y WOULD BE 2 TIMES 2 TIMES Y,
6Y WOULD BE 2 TIMES 3 TIMES Y,
25X SQUARED WOULD BE 5 TIMES 5 TIMES X TIMES X.
SO, NOW WE'LL IDENTIFY THE COMMON FACTORS
BETWEEN THE NUMERATORS AND DENOMINATORS.
SO, HERE WE HAVE 2 OVER 2.
HERE'S A Y OVER Y.
HERE'S A 5 OVER 5.
HERE, WE HAVE X OVER X, ANOTHER X OVER X.
LOOKS LIKE WE HAVE EVERYTHING.
LET'S GO AHEAD AND MULTIPLY NOW.
X TIMES 3 TIMES Y WOULD BE 3XY
AND OUR DENOMINATOR WOULD BE 2 TIMES 5, WHICH EQUALS 10.
LET'S GO AHEAD AND TAKE A LOOK AT ONE MORE EXAMPLE
OF MULTIPLICATION.
SO HERE, WE HAVE TO FACTOR EVERYTHING FIRST.
THESE ARE ALL TRINOMIALS WITH A LEADING COEFFICIENT OF 1,
SO THEY'D ALL FACTOR INTO TWO BINOMIALS IF THEY DO FACTOR.
SO, LOOKING AT THIS NUMERATOR,
WE'RE GOING TO HAVE Y IN THE FIRST POSITION.
THE FACTORS OF NEGATIVE 3 THAT ADD TO +2 PLUS 3 MINUS 1.
IN OUR DENOMINATOR, WE HAVE Y IN THE FIRST POSITION.
THE FACTORS ARE NEGATIVE 5 THAT ADD TO +4 + 5 - 1,
AND IN OUR SECOND FRACTION,
WE HAVE FACTORS OF Y AGAIN IN THE FIRST POSITION.
THE FACTORS OF NEGATIVE 10 THAT ADD TO NEGATIVE 3,
MINUS 5, PLUS 2.
AND THEN IN THIS LAST ONE,
THE FACTORS OF POSITIVE 6 THAT ADD TO POSITIVE 5,
PLUS 3, PLUS 2.
NOW, WE SIMPLIFY.
HERE, WE HAVE A Y MINUS 1 OVER Y MINUS 1.
HERE'S A Y PLUS 3 OVER Y PLUS 3 AND A Y PLUS 2 OVER A Y PLUS 2.
SO, NOW WE HAVE OUR PRODUCT IN SIMPLIFIED FORM.
WE HAVE Y MINUS 5 OVER Y PLUS 5.
AGAIN, BE CAREFUL HERE. WE CANNOT SIMPLIFY THIS.
THESE TERMS ARE ATTACHED BY ADDITION OR SUBTRACTION
AND IN ORDER TO SIMPLIFY,
THEY HAVE TO BE CONNECTED BY MULTIPLICATION.
OKAY, LET'S GO AHEAD AND TAKE A LOOK AT DIVISION NOW.
THE MAIN THING TO REMEMBER ABOUT DIVISION OF RATIONAL EXPRESSIONS
IS THAT WE CONVERT THEM TO MULTIPLICATION.
SO, INSTEAD OF DIVIDING BY W OVER Z,
WE WILL MULTIPLY BY THE RECIPROCAL
OR MULTIPLY BY Z OVER W.
ONCE WE CONVERT THE PROBLEM TO MULTIPLICATION,
WE WILL FOLLOW THE SAME STEPS ON THE PREVIOUS THREE EXAMPLES.
LET'S GO AHEAD AND TRY AN EXAMPLE.
SO, ON THIS PROBLEM, THE FIRST THING I NOTICE IS,
IT MIGHT BE HELPFUL TO HAVE THIS FIRST TERM IN FRACTION FORM.
NEXT, WE'LL GO AHEAD AND REWRITE THIS
AS A MULTIPLICATION PROBLEM.
SO, THIS IS THE SAME AS X SQUARED MINUS 5X,
MINUS 6 OVER 1,
TIMES THE RECIPROCAL OF THIS FRACTION.
SO, WE HAVE X PLUS 6 OVER X SQUARED MINUS 1.
NOW, WE'LL FACTOR EVERYTHING, SIMPLIFY, AND THEN MULTIPLY.
SO, FOR THIS FIRST NUMERATOR, WE'LL HAVE 2 BINOMIALS
WHERE THE FIRST TERM WILL BE X.
THE FACTORS OF NEGATIVE 6 THAT ADD TO NEGATIVE 5,
THAT'S MINUS 6 PLUS 1.
X PLUS 6 DOES NOT FACTOR.
X SQUARED MINUS 1 IS A DIFFERENCE OF SQUARED.
SO, WE HAVE X PLUS 1 TIMES X MINUS 1.
NOW, WE SIMPLIFY.
WE HAVE A FACTOR OF X PLUS 1 ON TOP AND X PLUS 1 ON THE BOTTOM.
THAT SIMPLIFIES.
SO, WE'RE LEFT WITH X MINUS 6, TIMES X PLUS 6,
ALL DIVIDED BY THE QUANTITY X MINUS 1.
I THINK WE HAVE TIME FOR ONE MORE EXAMPLE.
LET'S GO AHEAD AND CONVERT THIS
TO A MULTIPLICATION PROBLEM FIRST.
IF WE TAKE A LOOK AT THESE TERMS,
THIS NUMERATOR HERE LOOKS A LITTLE STRANGE
BECAUSE WE'RE USE TO HAVING THE VARIABLE FIRST
AND THE CONSTANT SECOND.
SO, IT MIGHT BE HELPFUL AT THIS TIME TO REARRANGE THESE TERMS.
IF WE REARRANGE THEM IN THEIR CURRENT FORM,
WE'D HAVE NEGATIVE Z CUBED PLUS ONE.
WE ALWAYS LIKE TO HAVE A POSITIVE LEADING COEFFICIENT
WHEN WE'RE FACTORING.
SO, WHAT WE CAN DO IS FACTOR OUT A NEGATIVE.
IF WE FACTOR OUT A NEGATIVE,
IT'S GOING TO CHANGE THE SIGN OF THIS Z CUBED.
SO, HERE IT'S MINUS Z CUBED.
IT WOULD BE POSITIVE Z CUBED
AND INSTEAD OF A POSITIVE 1, WE'D HAVE MINUS 1.
NOTICE IF WE DISTRIBUTE THE NEGATIVE SIGN
OR MULTIPLY IT BY A NEGATIVE 1,
IT DOES MATCH THE ORIGINAL NUMERATOR.
OVER Z SQUARED--
INSTEAD OF DIVISION, WE'LL MULTIPLY IT BY THE RECIPROCAL.
SO, WE HAVE Z TO THE FOURTH OVER Z SQUARED MINUS 1.
LET'S GO AHEAD AND FACTOR EVERYTHING NOW.
NOW, THIS IS A DIFFERENCE OF CUBES.
SO, THIS IS ONE OF THOSE FACTORING FORMULAS
THAT YOU MAY WANT TO REVIEW.
THE FIRST BINOMIAL WOULD BE Z MINUS 1.
THE NEXT FACTOR WILL BE A TRINOMIAL
WHERE WE HAVE Z SQUARED
AND THEN IT'S PLUS Z TIMES ONE, THAT'S PLUS Z,
PLUS 1 SQUARED WOULD BE 1.
TO FACTOR Z SQUARED, I'LL WRITE THAT AS Z TIMES Z.
Z TO THE FOURTH WOULD BE 4 FACTORS OF Z.
Z SQUARED MINUS 1 IS A DIFFERENCE OF SQUARES,
Z PLUS 1, Z MINUS 1.
NOW, WE SIMPLIFY.
THERE'S A Z MINUS 1 OVER A Z MINUS 1.
HERE, WE CAN SIMPLIFY OUT SOME FACTORS OF Z,
Z OVER Z AND Z OVER Z.
AND IT LOOKS LIKE WE HAVE EVERYTHING.
LET'S GO AHEAD AND MULTIPLY NOW.
YOU CAN SEE OUR FRACTION WILL BE NEGATIVE
AND OUR NUMERATOR IS GOING TO BE THIS TRINOMIAL TIMES Z SQUARED.
SO, WE'LL PUT THE Z SQUARED FIRST
AND OUR DENOMINATOR IS Z PLUS 1.
OKAY, THAT'S GOING TO DO IT FOR THIS VIDEO.
I HOPE YOU FOUND IT HELPFUL.
