- WELCOME TO AN INTRODUCTORY LESSON ON PARTIAL FRACTION
AND DECOMPOSITION.
IN THIS VIDEO, WE'LL PRACTICE SETTING UP THE FRACTIONS
THAT WILL BE USED TO PERFORM PARTIAL FRACTION DECOMPOSITION.
WE WON'T ACTUALLY PERFORM THE DECOMPOSITION.
PARTIAL FRACTION DECOMPOSITION IS A METHOD PROVIDING
A GIVEN RATIONAL EXPRESSION AS A SUM OR DIFFERENCE
OF SIMPLER FRACTIONS AND THIS IS OFTEN HELPFUL
WHEN PERFORMING INTEGRATION IN CALCULUS.
SO, LET'S FIRST REVIEW HOW WE'RE GOING TO SET THESE FRACTIONS UP.
FIRST, IF THE DEGREE OF THE NUMERATOR IS GREATER THAN
OR EQUAL TO THE DEGREE OF THE DENOMINATOR,
WE NEED TO PERFORM LONG DIVISION, BUT IN MOST CASES,
THIS WON'T BE NECESSARY.
SO, FOR STEP ONE, WE WANT TO START BY COMPLETELY
FACTORING THE DENOMINATOR INTO PRIME, LINEAR,
QUADRATIC FACTORS.
ONCE WE'VE DONE THIS, STEP TWO,
FOR EACH DISTINCT LINEAR FACTOR IN THE FORM AS WE SEE HERE,
WILL INCLUDE TERMS IN THE FORM
OF A DIVIDED BY THE LINEAR FACTOR.
FOR THE NEXT DISTINCT LINEAR FACTOR,
WE WOULD USE A NUMERATOR OF B, C, AND THEN SO ON.
IF WE HAVE REPEATED LINEAR FACTORS IN THE FORM,
THE QUANTITY PX PLUS 2, RAISED TO THE POWER OF N,
WE MUST INCLUDE N TERMS IN THE FORM THAT WE SEE HERE.
AGAIN, NOTICE HOW WE NEVER USE THE SAME VARIABLE
TWICE IN THE NUMERATOR.
THEN LOOKING AT THE DENOMINATOR,
NOTICE HOW WE START WITH THE SINGLE FACTOR
OR THE REPEATED FACTOR AND THEN,
WE PICK UP ONE ADDITIONAL FACTOR OF THE REPEATED FACTOR
IN SUCCESSIVE FRACTIONS UNTIL WE HAVE ALL OF THEM.
FOR STEP THREE, FOR EACH DISTINCT QUADRATIC FACTOR
OR FACTORS IN THIS FORM,
WE'LL INCLUDE FRACTIONS THAT LOOK LIKE THIS.
WHERE WE NOW NOTICE THE NUMERATOR IS IN THE FORM
BX PLUS C.
AND AGAIN, FOR ANY REPEATED QUADRATIC FACTOR
RAISED TO THE POWER OF N
WE MUST INCLUDE N TERMS THAT WOULD FIT THE FORM
THAT WE SEE HERE.
AGAIN, NOTICE HOW WE NEVER USE THE SAME VARIABLE TWICE
STARTING WITH A, B, C, D, E, AND SO ON AND THEN,
FOR THE REPEATED FACTORS,
WE START WITH A SINGLE FACTOR
AND THEN FOR EACH SUCCESSIVE FRACTION,
WE PICK UP AN ADDITIONAL FACTOR UNTIL WE HAVE ALL OF THEM.
AND THEN FINALLY STEP FOUR,
ONCE WE HAVE THE PARTIAL FRACTIONS,
WE'LL MULTIPLY EACH FRACTION BY THE LCD,
WHICH RESULTS IN WHAT'S CALLED A BASIC EQUATION,
WHICH WE'LL THEN SOLVE FOR A, B, C, D, AND SO ON.
SO, GOING BACK TO OUR THREE EXAMPLES,
NOTICE HOW THE DEGREE OF THE NUMERATOR
IS LESS THAN THE DEGREE OF THE DENOMINATOR.
SO, OUR FIRST STEP WILL BE TO FACTOR
THE DENOMINATOR COMPLETELY.
SO WE HAVE A COMMON FACTOR OF X SQUARED IN THE DENOMINATOR.
SO, WE'LL HAVE THE QUANTITY X PLUS 3 DIVIDED BY.
YOU WOULD HAVE X SQUARED TIMES X PLUS 9.
NOW, EVEN THOUGH X SQUARED IS DEGREE 2,
IT'S ACTUALLY THE LINEAR FACTOR X REPEATED TWICE.
SO, FOR OUR PARTIAL FRACTIONS,
WE'LL FIRST START WITH A DIVIDED BY ONE FACTOR OF X
AND THEN BECAUSE THE FACTOR OF X IS REPEATED,
THE NEXT FRACTION WOULD BE PLUS B DIVIDED BY X TO THE SECOND
AND THEN, WE STILL HAVE THE LINEAR FACTOR OF X PLUS 9.
SO, WE'LL HAVE PLUS C DIVIDED BY THE QUANTITY X PLUS 9.
THESE WOULD BE THE FRACTIONS THAT WE WOULD USE TO PERFORM
PARTIAL FRACTION DECOMPOSITION.
OKAY, LOOKING AT OUR SECOND EXAMPLE,
NOTICE HOW THE DENOMINATOR IS ALREADY IN FACTORED FORM
WHERE WE HAVE TWO DISTINCT LINEAR FACTORS
AND ONE DISTINCT QUADRATIC FACTOR.
STARTING WITH OUR LINEAR FACTORS,
WE WOULD HAVE A DIVIDED BY X,
PLUS B DIVIDED BY THE QUANTITY X MINUS 2
AND THEN FOR THE QUADRATIC FACTOR,
YOU'VE ALREADY USED A AND B, SO FOR THE NUMERATOR,
WE WOULD HAVE CX PLUS D.
NOTICE HOW WE'RE USING A LINEAR FACTOR IN THE NUMERATOR
BECAUSE WE HAVE A QUADRATIC FACTOR IN THE DENOMINATOR.
SO, WE'LL DIVIDE THIS BY THE QUANTITY X SQUARED PLUS 5
LET'S GO AND TAKE A LOOK AT ONE MORE EXAMPLE
THAT'S PRETTY INVOLVED.
AGAIN, NOTICE HOW THE DENOMINATOR
IS ALREADY IN FACTORED FORM,
BUT WE HAVE ONE DISTINCT LINEAR FACTOR,
TWO REPEATED LINEAR FACTORS,
AND THEN THREE REPEATED QUADRATIC FACTORS.
SO, WE'LL HAVE QUITE A FEW FRACTIONS HERE
SO LET'S START ON THE NEXT LINE.
WE'LL START WITH THE LINEAR FACTOR OF X.
SO, WE'LL HAVE A DIVIDED BY X
AND NOW WE'LL MOVE TO THE NEXT LINEAR FACTOR OF X MINUS 3,
WHICH WAS REPEATED.
SO, WE'LL NEXT HAVE PLUS B DIVIDED BY THE QUANTITY
X MINUS 3.
SO, WE'LL USE ONE FACTOR OF X MINUS 3 HERE
AND THE NEXT FRACTION, WE'LL USE TWO FACTOR OF X MINUS 3.
SO, WE'LL HAVE PLUS C DIVIDED BY THE QUANTITY X MINUS 3
TO THE SECOND.
AND NOW, WE'LL MOVE TO OUR QUADRATIC FACTORS
AND AGAIN, WE HAVE THREE FACTORS OR X SQUARED PLUS 2 SO PLUS.
NOW, WE'LL HAVE LINEAR FACTORS IN THE NUMERATOR.
SO, AFTER C, WE HAVE D, SO WE'LL HAVE DX PLUS E
DIVIDED BY ONE FACTOR OF X SQUARED PLUS 2.
NOW FOR NEXT FRACTION,
WE'LL PICK UP ANOTHER FACTOR OF X SQUARED PLUS 2
IN THE DENOMINATOR.
SO AFTER E, WE HAVE F.
SO, WE'LL HAVE FX PLUS G
AND THEN, WE'LL HAVE THE QUANTITY X SQUARED
PLUS 2 TO THE SECOND
AND BECAUSE WE HAVE THREE FACTORS OF X SQUARED PLUS 2,
WE'LL HAVE ONE MORE FRACTION.
AFTER G WOULD BE H,
SO WE'LL HAVE HX PLUS I DIVIDED BY THE QUANTITY X SQUARED PLUS 2
RAISED TO THE 3RD.
I'M GOING TO STOP HERE FOR THIS VIDEO.
IN THE NEXT SEVERAL VIDEOS,
WE'LL ACTUALLY BE PERFORMING THE PARTIAL FRACTION DECOMPOSITION.
I HOPE YOU FOUND THIS HELPFUL.