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THIS NEXT SECTION IS CONTINUING WITH OUR LETTERING.
LET'S CONTINUE WITH OUR LETTERING.
THE NEXT SECTION THAT WE WANT TO DO IN CHAPTER 7
IS TO FIND THE LEAST COMMON DENOMINATOR
WHICH WE CALL THE LCD.
OKAY, LET'S WRITE DOWN THE STEPS SO YOU WON'T FORGET.
FIRST THING YOU WANT TO DO
IS TO FACTOR EACH DENOMINATOR SEPARATELY.
SO REMEMBER, THE WHOLE STORY IS ABOUT DENOMINATORS,
SO FACTOR EACH DENOMINATOR SEPARATELY
AND THEN WHAT YOU NEED TO DO
IS TO GET ALL THE DIFFERENT FACTORS.
SO GET ALL THE DIFFERENT FACTORS,
THE DIFFERENT FACTORS,
AND TAKE THE HIGHEST EXPONENT IF POSSIBLE.
SO IF THERE'S A REPETITION, TAKE THE HIGHEST EXPONENT
AND WHAT HAPPENS IS THE LCD IS THE PRODUCT OF THESE FACTORS.
OKAY, SO LET'S TRY FROM OUR EXAMPLES
SO YOU UNDERSTAND WHAT WE'RE TALKING ABOUT.
OKAY, LET'S SAY THIS.
CAN YOU FIND THE LCD OF, SAY, 4X OVER X - 2,
5X + 1 OVER X SQUARED - 4X + 4
AND 3X CUBED - 2 OVER X SQUARED + 2X - 8?
SO REMEMBER WHAT THE D STANDS FOR.
D STANDS FOR DENOMINATOR,
OKAY, SO YOU DON'T CARE ABOUT THE NUMERATOR,
ONLY THE DENOMINATOR.
SO DO YOU NOTICE FOR CHAPTER 7,
ALTHOUGH WE'RE DEALING WITH RATIONAL EXPRESSION
FROM PRETTY MUCH THE BEGINNING OF CHAPTER 7,
BUT FOR DIFFERENT QUESTIONS,
DIFFERENT TOPICS IN CHAPTER 7,
YOU LOOK AT DIFFERENT THINGS.
SOMETIMES YOU LOOK AT BOTH,
SOMETIMES YOU ONLY LOOK AT THE DENOMINATOR.
SO NOW I'M FOCUSING ONLY AT THE BOTTOM,
SOMETHING LIKE THIS.
FOCUS ON THIS,
SO THE TRICK IS YOU'RE SUPPOSED TO FACTOR
EACH DENOMINATOR SEPARATELY.
SO X - 2, THERE'S NOTHING YOU CAN DO THERE.
BUT FOR X SQUARED - 4X + 4,
THE SUM IS A -4 AND THE PRODUCT IS A 4,
SO THE TWO NUMBERS ARE - 2 AND - 2.
SO IT'S GONNA BE X - 2 AND X - 2
AND YOU HAVE TO WRITE IT AS X - 2 SQUARED.
SO YOU WANT IT IN EXPONENT FORM.
LET'S DO THE LAST ONE.
THE SUM IS A +2 AND THE PRODUCT IS A -8.
WHICH TWO NUMBERS WILL DO THE TRICK?
+4, -2.
OKAY, SO LET'S TRY THAT.
X + 4, X - 2.
OKAY, SO WHAT I HAVE NOW IS INSTEAD OF THE DENOMINATORS
THAT WERE GIVEN TO ME ORIGINALLY,
I NOW HAVE X - 2,
X - 2 SQUARED,
AND X + 4, X - 2.
SO ACCORDING TO THE METHOD TO FIND THE LCD,
YOU ARE SUPPOSED TO FACTOR EACH DENOMINATOR SEPARATELY,
WHICH IS WHAT I DID. OBSERVE.
I TOTALLY IGNORED THE NUMERATOR, OKAY?
YET ALL THE DIFFERENT FACTORS,
NOW WHAT ARE THE DIFFERENT FACTORS?
SO THE DIFFERENT FACTORS ARE THE X - 2s AND THE X + 4.
THESE ARE THE DIFFERENT FACTORS
AND YOU'RE SUPPOSED TO TAKE THE HIGHEST EXPONENT IF POSSIBLE.
SO WHEN YOU SEE X - 2 BY ITSELF,
IT REALLY MEANS X - 2 TO POWER 1.
X - 2 SQUARED,
THAT MEANS HIS EXPONENT IS 2 VERSUS THE LAST ONE IS AGAIN 1.
SO OF THESE THREE EXPONENTS,
YOU'RE SUPPOSED TO PICK THE HIGHEST,
SO YOU WILL PICK X - 2 SQUARED,
AND YOU'RE SUPPOSED TO MULTIPLY THEM,
AND THERE YOU HAVE IT.
THAT'S THE LCD.
SO LCD IS WHERE YOU TAKE ALL THE DIFFERENT FACTORS
AND THE HIGHEST IF YOU HAVE A CHOICE.
SO YOU DON'T COMBINE THEM AND SAY X - 2 IS ABOUT 4.
YOU JUST TAKE THE HIGHEST OF THE LOT.
SO THIS ACTUALLY IS NOT TOO MUCH DIFFERENT
FROM WHAT YOU HAVE LEARNED BEFORE.
SO IF YOU RECALL, LET'S GO BACK TO FAMILIAR GROUNDS.
LET'S DEAL WITH FRACTIONS, OKAY?
FOR EXAMPLE, IF I SAY, CAN YOU FIND THE LCD OF 1/2,
5/8 AND, LET'S SAY,
I DON'T KNOW,
3/16, FOR EXAMPLE, OKAY?
SO YOU NOTICE,
YOU DON'T CARE ABOUT THE 1 AND THE 5 AND THE 3.
ALL YOU CARE ABOUT IS THE 2, 8, 16
'CAUSE WE'RE LOOKING FOR LCD.
THEN YOU ARE THINKING THAT 8 IS REALLY 2 x 2 x 2 THREE TIMES,
SO IT'S 2 CUBED.
THIS IS 2 TO 1.
AND 16 IS REALLY 2, 2, 2, 2, SO IT'S 2 TO THE POWER OF 4.
SO OF THESE THREE, YOU WILL SAY THE LCD IS 16
BECAUSE, BETWEEN 2, 1, 2, 3 AND 2 TO THE 4TH POWER,
2 TO THE 4TH POWER IS THE HIGHEST OF THE LOT.
IF THEY'RE ALL 2s, BUT DIFFERENT EXPONENTS,
SO WE TAKE THE HIGHEST ONE.
SO THIS IS EXACTLY WHAT WE'RE DOING HERE.
WE ARE TAKING THE HIGHEST EXPONENT WHERE WE HAVE A CHOICE.
SO IN THIS LOT OF FRACTIONS,
1/2, 5/8 AND 3/16,
WE WILL SAY THE LCD IS 16.
WE DON'T MULTIPLY ALL THESE NUMBERS TOGETHER.
SO THIS IS THE SAME CONCEPT THAT YOU HAVE LEARNED BEFORE
EXCEPT THAT NOW WE ARE DEALING WITH ALGEBRA.
OKAY, LET'S TRY ANOTHER EXAMPLE.
LET'S CALL THIS EXAMPLE TWO.
CAN YOU FIND THE LCD OF, LET'S SEE,
1 OVER X SQUARED + 6X + 8,
X OVER X SQUARED - 16
AND THEN LET'S TRY 4X + 3 OVER X CUBED + 8X SQUARED + 16X.
OKAY, THAT'S TRY THAT.
OKAY, JUST FOCUSING AT THE DENOMINATORS, ALL RIGHT.
FACTOR EACH ONE.
SO THE FIRST ONE IS A TRINOMIAL.
SUM IS 6, PRODUCT IS 8, SO I'M LOOKING AT 4 AND 2,
SO X + 4, X + 2.
OKAY, LET'S LOOK AT X SQUARED - 16.
THAT'S A DIFFERENCE OF SQUARES, SO IT'S X AND X,
4 AND 4, 1 PLUS, 1 MINUS.
ALL RIGHT, LET'S TRY THIS ONE.
I NEED A GCF OF X FIRST FOLLOWED BY X SQUARED + 8X + 16.
OKAY, I HAVE A SUM OF 8 AND A PRODUCT OF 16,
SO THE TWO NUMBERS THAT WILL WORK IS 4 AND 4,
SO WHAT IS THE ANSWER FOR MY LAST DENOMINATOR?
LET'S EXTEND THE ARROW.
IT'S GOING TO BE X TIMES OF X + 4, X + 4,
SO WE'RE GONNA CALL IT X + 4 SQUARED.
OKAY, SO WHAT DO WE HAVE HERE?
WE HAVE--SO WHAT'S THE LCD?
OKAY, WE NEED ALL THE DIFFERENT FACTORS,
SO THIS X + 4 HERE AND HERE AND HERE,
BUT BETWEEN THEM I NEED THE HIGHEST EXPONENT,
SO IT'S GONNA BE X + 4 SQUARED.
WHAT ELSE DO I HAVE?
I HAVE X + 2.
THERE'S NO MORE X + 2s, BUT I STILL TAKE IT ANYWAY
'CAUSE WE NEED ALL THE DIFFERENT ONES,
ALL NEW DIFFERENT FACTORS.
WHAT ELSE? X - 4.
HE'S THE ONLY ONE WHO HAS IT,
BUT WE'LL TAKE IT ANYWAY,
AND THE LAST ONE, X RIGHT HERE, SO WE TAKE THAT, TOO.
SO LCD IS REALLY ALL THE DIFFERENT FACTORS,
OKAY, AND THE HIGHEST EXPONENT IF YOU HAVE A CHOICE.
OKAY, SO LET'S CALL THIS EXAMPLE THREE,
BUT EXAMPLE THREE IS A BIT DIFFERENT
FROM EXAMPLE ONE AND TWO
BECAUSE I'M GONNA GET YOU TO DO A FEW THINGS.
FIRST AND FOREMOST, I NEED YOU TO FIND THE LCD
AND I NEED YOU TO WRITE OR REWRITE
THE RATIONAL EXPRESSIONS
WITH THE LCD AS THE NEW DENOMINATOR.
OKAY, WHAT DOES IT MEAN
BY REWRITE THE RATIONAL EXPRESSION WITH THE LCD
AS THE NEW DENOMINATOR?
SO THIS IS SOMETHING, AGAIN, YOU HAVE SEEN BEFORE.
FOR EXAMPLE, THE FRACTION IS, LET'S SAY, 2/3.
SAY I NO LONGER WANT YOU TO USE 3 AS THE DENOMINATOR.
I SAY, OKAY, CAN YOU USE 45 AS THE DENOMINATOR?
THEN THAT'S WHEN YOU HAVE TO THINK,
OKAY, I HAVE A 3 RIGHT NOW.
THAT'S RIGHT NOW WHAT I HAVE.
HOW DO I MAKE 3 BECOME A 45?
SO YOU HAVE TO THINK HOW DO I HAVE TO MULTIPLY?
SO THE KEY IS YOU NEED TO MULTIPLY BY SOMETHING
AND I NEED TO USE 15.
3 x 15 WILL GIVE ME 45,
BUT AT THE SAME TIME,
WHEN I MULTIPLY THE DENOMINATOR BY 15,
I WILL HAVE TO MULTIPLY THE NUMERATOR BY 15 AS WELL.
SO THE ORIGINAL NUMERATOR WAS A 2,
BUT NOW WHEN I MULTIPLY BY 15, IT TURNS IT INTO A 30.
BUT THE THING IS, IF YOU REDUCE THIS,
YOU WILL STILL GO BACK TO YOUR 2/3.
SO THIS IS REALLY WHAT WE USE TO CALL EQUIVALENT FRACTIONS
AS IN LIKE FRACTIONS THAT LOOK DIFFERENT,
BUT REALLY THEIR VALUES ARE THE SAME.
AND THIS IS REALLY WHAT THEY ARE TRYING TO GET YOU TO DO.
FIRST, YOU FIND THE LCD
AND THEY WANT YOU TO REWRITE EVERY EXPRESSION
USING THE LCD AS THE NEW DENOMINATOR.
SO EQUIVALENTLY, IT'S LIKE ASKING YOU
CAN YOU CONVERT 2/3 TO A NEW FRACTION
WITH 45 AS THE DENOMINATOR?
SOMETHING LIKE THAT, OKAY?
NOW LET'S TRY THE EXAMPLE, AND LET'S SAY YOU'RE GIVEN...
LET'S SAY YOU ARE GIVEN THE FRACTIONS X OVER X CUBED + 27
AND -2 OVER X SQUARED + 6X + 9.
LET'S TRY THAT.
OKAY, SO LET'S LOOK AT THE DENOMINATORS.
RIGHT NOW I HAVE X CUBED + 27
AND THAT'S SUPPOSED TO BE CUBED,
SO IT'S X CUBED + 3 CUBED,
SO A LITTLE PARENTHESES, BIG ONE,
X + 3 IS IN THE LITTLE ONE.
X SQUARED, 3X AND 9.
I NEED OPPOSITE SIGNS FOR THE FIRST ONE
AND THE SECOND ONE I NEED IT TO BE A PLUS.
OKAY, AND FOR X SQUARED + 6X + 9.
SO X CUBED + 27 IS X + 3,
X SQUARED + 3X + 9.
SORRY, THERE'S A MINUS RIGHT THERE.
X SQUARED + 6X + 9.
THAT'S A TRINOMIAL, SUM IS 6, PRODUCT IS 9,
SO THE TWO NUMBERS IS 3 AND 3,
SO THAT'S GONNA BE X + 3 x X + 3,
WHICH IS X + 3 SQUARED.
SO LET'S CALL IT X + 3 SQUARED.
SO WHAT'S THE LCD IN THIS CASE?
BETWEEN X + 3 AND X + 3 SQUARED,
I NEED TO USE X + 3 SQUARED AND I NEED ALL THE FACTORS,
SO EVEN THE X SQUARED + 3X + 9
HAS TO BE CONSIDERED IN THE LCD AND THAT'S IT.
I THINK WE TOOK CARE OF EVERYBODY THAT BECAME MY LCD.
SO THAT'S THE FIRST PART OF THE QUESTION CAN YOU FIND THE LCD.
SO THE LCD IS DONE, FINISHED.
OKAY, NEXT PART OF THE QUESTION IS CAN YOU REWRITE
EACH EXPRESSION USING LCD AS THE NEW DENOMINATOR?
NOW THIS IS HOW YOU HAVE TO DO IT.
THE FIRST FRACTION GIVEN TO YOU IS X OVER X CUBED + 27.
HIS DENOMINATOR IS X OVER X + 3,
X SQUARED - 3X + 9.
BUT YOU WANT THE NEW DENOMINATOR TO BE X + 3 SQUARED
AND THEN X SQUARED - 3X + 9.
NOW THIS IS WHAT YOU NEED TO OBSERVE.
WHEN THE NUMERATOR IS X, HIS DENOMINATOR IS X + 3, THERE,
X + 3, X SQUARED - 3X + 9, LIKE THAT.
SO YOU NEED TO COMPARE THIS DENOMINATOR
WITH THE NEW DENOMINATOR.
THIS IS OLD DENOMINATOR,
OKAY, AND THAT'S NEW DENOMINATOR.
SO BETWEEN THE OLD AND THE NEW DENOMINATOR.
BETWEEN THE OLD AND THE NEW, WHAT IS EXTRA?
SO YOU NOTICE THAT, HEY, THERE'S AN EXTRA X + 3.
SO THAT TELLS YOU YOU HAVE TO GO OVER HERE TO MULTIPLY BY X + 3.
REMEMBER THE EXAMPLE ON NUMBERS,
OKAY? THIS IS EXACTLY HOW YOU HAVE TO THINK.
SO YOU NOTICE I'M MISSING AN X + 3 FOR THE BOTTOM
AND THAT MEANS I HAVE TO MULTIPLY AN X + 3 FOR THE TOP.
SO WHAT YOU DO TO THE BOTTOM, JUST DO TO THE TOP AS WELL.
AND SO NOW THE NEW NUMERATOR IS NO LONGER X
BECAUSE I NEED TO DISTRIBUTE THIS.
SO WHAT I HAVE,
THE NEW NUMERATOR IS GOING TO NOW BE X SQUARED + 3X
AND THAT'S ALL THEY WANT.
OKAY, THEY WANT YOU TO REWRITE
THE OLD FRACTION WITH THE LCD AS THE NEW DENOMINATOR,
BUT AT THE SAME TIME WHEN YOU CHANGE THE DENOMINATOR,
YOU MIGHT HAVE TO CHANGE THE NUMERATOR IF THEY'RE DIFFERENT.
OKAY, LET'S TRY THE OTHER ONE. WHICH ONE IS THIS?
-2 OVER X SQUARED + 6X + 9.
SO HIS DENOMINATOR IS X + 3 SQUARED,
BUT WHAT I WANT IS X + 3 SQUARED x X SQUARED - 3X + 9.
AGAIN, COMPARE THE TWO.
YOU NOTICE, HEY, YOU'RE MISSING AN X SQUARED - 3X + 9.
SO THAT MEANS YOU HAVE TO MULTIPLY THE TOP AS WELL
BY X SQUARED - 3X + 9.
SO WHEN I DISTRIBUTE, -2 INTO THE NEW NUMERATOR,
IT NOW READS -2X SQUARED + 6X - 18.
OKAY, LET'S CALL THIS EXAMPLE FOUR
AND LET'S DO IT VERY SIMILAR TO EXAMPLE THREE, SAME THING.
CAN YOU FIND THE LCD
AND REWRITE EACH RATIONAL EXPRESSION
WITH THE LCD AS THE NEW DENOMINATOR?
LET'S TRY THAT.
LET'S SEE, LET'S DO ONE THAT HAS THREE FRACTIONS.
HOW'S THAT? OKAY?
NOW TRY THIS.
1 OVER X + 1, X OVER X - 2,
X SQUARED + 2 OVER X SQUARED - X - 2.
LET'S TRY THAT.
SO AS FAR AS DENOMINATORS ARE CONCERNED,
THERE'S NOT MUCH I CAN DO WITH X + 1.
THERE'S NOT MUCH I CAN DO WITH X - 2,
SO THAT'S AS FACTORED AS I CAN GET.
BUT THE THIRD ONE, THE SUM IS A -1
AND THE PRODUCT IS A -2.
SO WHICH TWO NUMBERS WILL DO THE TRICK?
I'M GOING TO USE -2 AND +1.
SO WHAT I HAVE IS X - 2 AND X + 1.
OKAY, SO LET'S DETERMINE THE LCD.
SO X + 1, I'M GOING DOWN THE LINE.
I FOUND ANOTHER X + 1 DOWN THERE.
THEY ARE REALLY RAISED TO THE SAME EXPONENT.
REMEMBER, YOU GET TO PICK THE HIGHER OF THE TWO
AND THEY'RE THE SAME, SO LET'S JUST CALL IT X + 1.
GOING DOWN THE LIST,
I HAVE X - 2 AND DOWN THE LIST IS ANOTHER X - 2 AGAIN.
THESE TWO ARE THE SAME EXPONENT.
THEY ARE RAISED TO EXPONENT 1, SO YOU GET TO PICK.
YOU'RE NOT GONNA GET TO COMBINE,
OKAY? YOU DON'T COMBINE THEM.
YOU JUST PICK THE HIGHER OF THE TWO.
SINCE THEY ARE THE SAME, 1 AND 1, RIGHT?
SO WE'LL JUST PICK ONE.
SO YOU MULTIPLY THESE AND THAT'S MY LCD THERE.
DONE.
OKAY, NOW THE NEXT PART OF THE QUESTION
IS CAN YOU REWRITE EACH FRACTION
USING THE LCD AS THE NEW DENOMINATOR?
SO LET'S GO THIS WAY.
SO 1 OVER X + 1. THAT WAS THE ORIGINAL.
AND WE WANT THE NEW DENOMINATOR TO BE LCD,
MEANING WE WANT IT TO BE THIS.
SO YOUR JOB IS TO FIGURE OUT WHAT'S ON THE TOP.
SO GO TO THE OLD DENOMINATOR AND COMPARE WITH THE NEW,
SO OLD AND NEW.
JUST KEEP COMPARING THESE TWO LIKE THAT.
YOU NOTICE THAT YOU ARE MISSING AN X - 2.
THAT'S WHAT YOU'RE MISSING.
SO I NEED TO MULTIPLY AN X - 2 FOR THE BOTTOM,
THAT'S THE DENOMINATOR, TO MAKE IT MATCH, RIGHT?
SO IF I DO THAT TO THE DENOMINATOR,
THEN MAKE SURE YOU DO THAT TO THE NUMERATOR AS WELL.
SO 1 x X - 2 IS X - 2. NOW WATCH OUT.
DON'T CANCEL THE X - 2 BECAUSE, IF YOU DO THAT,
YOU GO BACK TO WHERE YOU CAME FROM
AND THAT'S NOT WHAT THE QUESTION IS ASKING YOU TO DO.
THEY WANT YOU TO REWRITE EACH FRACTION
USING THE LCD AS THE NEW DENOMINATOR,
SO THIS IS THE ANSWER WE WANT
BECAUSE THEY HAVE THE LCD AS THE DENOMINATOR.
BECAUSE IF YOU TRY AND SIMPLIFY IT
AS IN YOU CANCEL THE X - 2 LIKE THIS,
YOU WILL NOTICE THAT YOU'RE BACK TO WHERE YOU CAME FROM
AND YOU DON'T WANT THAT.
SO TRY THE NEXT ONE, X OVER X - 2.
AGAIN, YOU WANT THE X + 1 AND X - 2 AS THE NEW DENOMINATOR.
SO AGAIN, COMPARE.
OKAY, THAT'S ALWAYS THE CASE, COMPARE.
SO COMPARE,
SAME HERE, COMPARE.
YOU NOTICE YOU ARE MISSING THE X + 1.
SO THAT MEANS I HAVE TO DO THE X + 1 FOR THE TOP AS WELL.
THIS IS MULTIPLICATION, SO I HAVE TO DISTRIBUTE.
SO YOU'RE NO LONGER AN X.
YOU'RE NOW X SQUARED + X, AND THAT'S IT.
THAT'S WHAT WE WANT.
THIS IS ACTUALLY TO PREPARE YOU FOR THE NEXT SECTION
WHERE WE ARE DEALING
WITH ADDITION AND SUBTRACTION OF FRACTIONS.
SO WHAT YOU'RE DOING RIGHT NOW,
IT DOESN'T QUITE MAKE SENSE TO YOU
BECAUSE YOU'LL BE WONDERING, WHY AM I DOING THIS?
WHY AM I MAKING THE FRACTIONS SO BIG?
BUT THAT'S REALLY TO PREPARE YOU FOR THE NEXT SECTION, OKAY?
ALL RIGHT, LAST ONE.
X SQUARED + 2 OVER X SQUARED - X - 2.
SO THIS ONE, THE DENOMINATOR WAS FACTORED,
AS X - 2 AND X + 1, RIGHT?
YOU WANT THE DENOMINATOR TO BE X - 2 AND X + 1
EXACTLY AS YOU SEE IT.
SO YOU NOTICE THAT,
IF YOU COMPARE THE TWO, COMPARE AGAIN,
THERE IS NOTHING MISSING.
SO NOTHING MISSING AS IN YOU TIMES 1.
SO IF YOU TIMES 1 IT'S AS GOOD AS YOU DON'T DO ANYTHING TO IT,
SO THE NUMERATOR STAYS.
SO IF YOU DON'T NEED TO ADD ANYTHING TO THE DENOMINATOR,
THEN YOU DON'T NEED TO ADD ANYTHING TO THE NUMERATOR,
ALL RIGHT, AND THAT'S THAT.
OKAY, CAN YOU TRY SOME AS WELL?
THIS IS FOUND ON PAGE 319,
OKAY? LET'S GET YOU TO DO THESE.
OKAY, 23, 25, 31 AND 47. �