- WELCOME TO A VIDEO
ON SIMPLIFYING RATIONAL EXPRESSIONS.
FIRST, LET'S DEFINE A RATIONAL EXPRESSION.
IF WE LET U AND V BE POLYNOMIALS,
THEN THE EXPRESSION, U DIVIDED BY V,
USUALLY WRITTEN IN FRACTION FORM IS A RATIONAL EXPRESSION.
AND WE SHOULD NOTE THAT THE DOMAIN
OF THIS RATIONAL EXPRESSION IS A SET OF ALL REAL NUMBERS,
FOR WHICH THE DENOMINATOR DOES NOT EQUAL ZERO.
SO, HERE ARE SOME EXAMPLES OF RATIONAL EXPRESSIONS.
YOU CAN SEE TWO-THIRDS IS A RATIONAL EXPRESSION.
SO, WE'VE DEALT WITH THESE WAY BACK IN ARITHMETIC,
BUT NOW SINCE THE NUMERATOR AND DENOMINATOR CAN BE POLYNOMIALS,
THEY CAN TAKE ON A VARIETY OF FORMS.
SO, NOTICE M DIVIDED BY N IS ONLY DEFINED
WHEN N DOES NOT EQUAL ZERO.
AND FOR THIS RATIONAL EXPRESSION,
THIS IS ONLY DEFINED WHEN X DOES NOT EQUAL H,
OKAY LET'S TAKE A LOOK AT
SIMPLIFYING RATIONAL EXPRESSIONS.
IF WE HAVE A RATIONAL EXPRESSION IN THE FORM OF U TIMES W,
DIVIDED BY V TIMES W,
WE CAN SEE THERE'S A COMMON FACTOR OF W
BETWEEN THE NUMERATOR AND DENOMINATOR
AND WE KNOW THAT W DIVIDED BY W WOULD EQUAL 1,
SO THIS W SIMPLIFIES OUT
AND WE'RE LEFT WITH THE SIMPLIFIED RATIONAL EXPRESSION
U/V.
SO, TO SIMPLIFY A RATIONAL EXPRESSION,
WHAT WE'RE GOING TO DO
IS FACTOR THE NUMERATOR AND DENOMINATOR COMPLETELY
AND THEN SIMPLIFY OUT THE COMMON FACTORS.
SO ONCE AGAIN, WHEN SIMPLIFYING RATIONAL EXPRESSIONS,
WE NEED TO BE AWARE THAT THE SIMPLIFIED EXPRESSION
IS ONLY DEFINED WHEN THE ORIGINAL EXPRESSION IS DEFINED.
SO, HERE'S AN EXAMPLE WHERE THIS RATIONAL EXPRESSION
DOES SIMPLIFY BECAUSE WE HAVE AN X - 1/X - 1.
SO, IT SIMPLIFIES TO X + 2, DIVIDED BY X + 5.
SO, THIS IS ONLY DEFINED WHEN X DOES NOT EQUAL -5,
BUT X ALSO CAN'T EQUAL 1
BECAUSE 1 WOULD MAKE THE ORIGINAL RATIONAL EXPRESSION
UNDEFINED.
LET'S GO AND TAKE A LOOK AT SOME ADDITIONAL EXAMPLES.
SO, WE'LL SIMPLIFY ALL OF THESE
BY WRITING THEM OUT IN FACTORED FORM AND THEN LOOKING
FOR THE COMMON FACTORS.
SO, WE KNOW THERE ARE SOME SHORTCUTS
TO SIMPLIFYING NUMBER ONE.
I'M GOING TO GO AHEAD AND WRITE THIS OUT IN PRIME FACTORED FORM.
SO, I'LL WRITE 10 AS 2 x 5
X TO THE THIRD WOULD BE 3 FACTORS OF X.
Y SQUARED WOULD BE 2 FACTORS OF Y, 25 IS 5 x 5.
X SQUARED, Y SQUARED.
SO NOW, WE'LL LOOK FOR THE COMMON FACTORS
BETWEEN THE NUMERATOR AND DENOMINATOR AND HERE,
WE HAVE 5 OVER 5, X/X, X/X, Y/Y, AND Y/Y.
SO, WHATEVER WE HAVE NOT SIMPLIFIED OUT
WOULD BE OUR SIMPLIFIED RATIONAL EXPRESSION.
LOOKS LIKE WE HAVE 2X IN THE NUMERATOR
AND 5 IN THE DENOMINATOR.
AND WE SHOULD ALSO MAKE A NOTE THAT X CAN'T EQUAL ZERO
AND Y CAN'T EQUAL ZERO AS WELL
BECAUSE THE ORIGINAL RATIONAL EXPRESSION WOULD BE UNDEFINED
IF X AND Y DID TAKE ON THOSE VALUES.
OKAY, ON NUMBER TWO, WE NEED TO BE CAREFUL HERE.
IT MAY BE TEMPTING TO SIMPLIFY THIS 3 AND THIS 6,
BUT WE CAN'T SIMPLIFY ACROSS ADDITION.
SO, WHAT WE SHOULD DO IS FACTOR
THE NUMERATOR AND DENOMINATOR COMPLETELY.
SO, LOOKING AT THE NUMERATOR,
THERE'S A COMMON FACTOR OF 3Y BETWEEN THESE TWO TERMS.
IF WE FACTOR OUT 3Y, WE'RE LEFT WITH Y + 4
AND IN THE DENOMINATOR, WE HAVE 6Y,
WHICH CAN BE WRITTEN AS 2 x 3 x Y.
AGAIN, WE CAN SIMPLIFY OUT FACTORS
THAT ARE MULTIPLIED TOGETHER.
SO, NOTICE WE HAVE A COMMON FACTOR OF 3
AND ALSO A COMMON FACTOR OF Y.
SO, OUR NUMERATOR SIMPLIFIES TO Y + 4
AND OUR DENOMINATOR SIMPLIFIES TO 2.
AGAIN, IT'S VERY TEMPTING TO SIMPLIFY HERE,
BUT WE CAN'T DO THAT BECAUSE THESE ARE ATTACHED BY ADDITION.
AND AGAIN, WE MAY WANT TO MAKE A NOTE
THAT Y CAN'T EQUAL ZERO
BECAUSE THAT WOULD MAKE THE ORIGINAL RATIONAL EXPRESSION
UNDEFINED.
LET'S TAKE A LOOK AT A FEW MORE.
AGAIN, IT MAY BE TEMPTING TO SIMPLIFY THESE X SQUAREDS OUT,
BUT WE CANNOT DO THAT BECAUSE THESE TERMS
ARE ATTACHED TO OTHER TERMS BY ADDITION OR SUBTRACTION.
SO, WE'LL FACTOR THIS, IT'S A DIFFERENCE OF SQUARES.
WE'D HAVE X + 3 x X - 3.
AGAIN, IF YOU'RE GOOD AT FACTORING,
YOU'LL BE GOOD AT SIMPLIFYING THESE TYPES OF EXPRESSION.
HERE IN THE FIRST POSITION,
WE'LL HAVE THE FACTORS OF X SQUARED, THAT'S X AND X.
NOW, WE WANT THE FACTORS OF -15 THAT ADD TO -2.
THAT WOULD BE -5 AND 3.
NOW, WE CAN SEE WE HAVE A COMMON FACTOR OF X + 3.
SO, THAT SIMPLIFIES TO X - 3/X -5
AND JUST TO BE CONSISTENT,
WE KNOW THAT X CAN'T EQUAL 5
AND X CAN'T EQUAL -3 BECAUSE IT WOULD MAKE THIS UNDEFINED.
MOVING ONTO NUMBER FOUR, TYPICALLY,
THE NUMERATOR HERE WOULD BE MORE DIFFICULT TO FACTOR
SINCE THE VALUE OF A IS NOT EQUAL TO 1.
SO, LET'S GO AHEAD AND FACTOR THE DENOMINATOR FIRST
BECAUSE THAT MAY HELP US WITH THE NUMERATOR.
SO, WE HAVE X IN THE FIRST POSITION.
WE WANT THE FACTORS OF POSITIVE 12 THAT ADD TO -7, -4 AND -3.
NOW, WE KNOW WE'RE PROBABLY GOING TO HAVE A COMMON FACTOR
IN THE NUMERATORS.
LET'S GO AHEAD AND TRY TO USE THAT INFORMATION
TO HELP US HERE.
WE KNOW THAT IF THIS FIRST TERM IS 3X SQUARED,
WE HAVE TO HAVE A 3X AND X
AND WE KNOW THE SECOND POSITIONS
MUST COME FROM THE FACTORS OF -8
SINCE I SEE A - 4 HERE,
I'M GOING TO TRY PUTTING THE - 4 HERE,
WHICH WOULD MAKE THE OTHER FACTOR A 2.
LET'S GO AHEAD AND CHECK THIS. WE HAVE -12X, 2X.
THAT DOES SUM TO -10X.
SO, WE'VE FACTORED CORRECTLY
AND NOW WE CAN SEE THERE'S A COMMON FACTOR OF X - 4.
SO, THIS SIMPLIFIES TO 3X + 2/X - 3
WHERE X CAN'T EQUAL 4 AND X CAN'T EQUAL 3.
OKAY, LET'S TAKE A LOOK AT ONE MORE.
LOOKING AT THIS, WE SEE OUR NUMERATOR DOES NOT FACTOR.
SO, WE'LL LEAVE THIS AS X - 7
NOW, THERE IS A COMMON FACTOR OF 3 IN THE DENOMINATOR.
LET'S GO AHEAD AND FACTOR THAT OUT.
WE FACTOR OUT 3, WE'RE LEFT WITH 7 - X.
NOW, THIS LOOKS A LITTLE BIT DIFFERENT
BECAUSE THESE TERMS ARE THE SAME,
BUT THEY'RE THE OPPOSITE SIGN.
HERE, WE HAVE A X AND A - 7.
HERE, WE HAVE A 7 AND A - X.
BUT IF WE WERE TO FACTOR OUT A -3 INSTEAD OF A 3,
IT WOULD CHANGE THE SIGN OF THESE TWO TERMS
AND THEREFORE, MATCH THE FACTOR IN THE NUMERATOR.
LET'S GO AHEAD AND TRY THAT.
IF WE FACTOR OUT -3,
THIS X WOULD BECOME POSITIVE AND THIS 7 WOULD BECOME NEGATIVE,
SO WE CAN WRITE X - 7.
LET'S CHECK IT.
IT'S A NEGATIVE 3X + 21,
WHICH DOES MATCH THE ORIGINAL DENOMINATOR.
SO, NOW WE CAN SEE THAT WE DO HAVE A COMMON FACTOR
OF X - 7 IN THE NUMERATOR AND DENOMINATOR.
SO, THIS SIMPLIFIES TO A 1
IN THE NUMERATOR AND NEGATIVE 3 IN THE DENOMINATOR,
MORE TYPICALLY WRITTEN AS -1/3.
AND JUST TO BE CONSISTENT, WE KNOW THAT X CANNOT EQUAL 7
BECAUSE THAT WOULD MAKE OUR DENOMINATOR ZERO.
OKAY, I HOPE YOU FOUND THIS VIDEO HELPFUL.
THANK YOU AND HAVE A GOOD DAY.