- WELCOME TO ANOTHER EXAMPLE OF FINDING THE PARTIAL FRACTION
DECOMPOSITION OF A GIVEN RATIONAL EXPRESSION.
NOTICE THE DEGREE OF THE NUMERATOR
IS < THE DEGREE OF THE DENOMINATOR.
SO WE CAN GO AHEAD AND PROCEED
BY FIRST FACTORING THE DENOMINATOR COMPLETELY.
WELL, X TO THE 3rd + 4X SQUARED
HAS A COMMON FACTOR OF X SQUARED.
SO WE'D HAVE THE QUANTITY X SQUARED + 8 DIVIDED BY X SQUARED
x THE QUANTITY X + 4.
NOTICE IN THIS FORM WE HAVE 3 LINEAR FACTORS,
THOUGH THE LINEAR FACTOR OF X IS REPEATED
WHICH WILL AFFECT HOW WE SET UP OUR FRACTIONS ON THE RIGHT SIDE.
AGAIN BECAUSE WE HAVE 2 LINEAR FACTORS OF X
OUR FIRST FRACTION WILL HAVE A DENOMINATOR OF 1 FACTOR OF X.
THE SECOND FRACTION WILL HAVE A DENOMINATOR OF X SQUARED.
THIS IS HOW WE HANDLE IT IF WE HAVE A REPEATED LINEAR FACTOR
PLUS THE 3rd LINEAR FACTOR OF X + 4.
AGAIN BECAUSE WE HAVE LINEAR FACTORS IN THE DENOMINATOR,
THAT MEANS OUR FACTORS WILL BE CONSTANT
BUT WE DON'T KNOW THOSE VALUES RIGHT NOW.
SO WE'RE GOING TO USE THE VARIABLES A, B, AND C.
SO OUR GOAL HERE IS TO FIND THE VALUES OF A, B, AND C
SO THAT WE KNOW THE 3 FRACTIONS THAT IF WE SUM THEM
WE OBTAIN THE ORIGINAL RATIONAL EXPRESSION.
SO FOR THE NEXT STEP WE'RE GOING TO CLEAR THE FRACTIONS
FROM THIS EQUATION BY MULTIPLYING
BOTH SIDES OF THE EQUATION BY THE LCD,
WHICH WOULD BE X SQUARED x THE QUANTITY X + 4.
SO NOTICE ON THE LEFT SIDE EVERYTHING SIMPLIFIES NICELY.
X SQUARED/X SQUARED, X + 4/X + 4.
SO WE JUST HAVE X SQUARED + 8.
ON THE RIGHT SIDE FOR THE FIRST PRODUCT ONLY ONE FACTOR OF X
SIMPLIFIES OUT.
THAT WOULD GIVE US AX x THE QUANTITY X + 4.
FOR THE NEXT PRODUCT X SQUARED/X SQUARED SIMPLIFIES OUT
LEAVING US WITH B x THE QUANTITY X + 4.
FOR THE 3rd PRODUCT WE HAVE X + 4/X + 4,
THAT SIMPLIFIES TO CX SQUARED.
THIS IS CALLED OUR BASIC EQUATION,
WHICH WE'LL USE TO FIND THE VALUES OF A, B, AND C.
AND THERE ARE 2 METHODS TO DO THIS;
1 IS TO SELECT THE VALUES OF X
THAT WOULD MAKE THE FACTORS INVOLVING X = 0,
WHICH WE'VE DONE ON THE PREVIOUS EXAMPLES.
THE OTHER METHOD IS TO USE A METHOD CALLED
EQUATING THE COEFFICIENTS, WHICH WE'LL DO IN THIS EXAMPLE.
SO WE'RE GOING TO MULTIPLY OUT THE RIGHT SIDE
AND THEN COMBINE LIKE TERMS BY FACTORING.
SO WE'LL HAVE X SQUARED + 8 = HERE WE'D HAVE AX SQUARED
+ 4AX + BX + 4B + CX SQUARED.
NOW WE WANT TO GROUP THE X SQUARED TERMS,
THE X TERMS AND THE CONSTANTS.
SO WE'D HAVE X SQUARED + 8 EQUALS
WE'D HAVE AX SQUARED + CX SQUARED.
NOTICE HOW WE HAVE THE X SQUARE TERMS NEXT TO EACH OTHER + 4AX
+ BX + 4B.
AND NOW WE'LL FACTOR X SQUARED OUT OF THESE TWO TERMS
AND X OUT OF THESE TWO TERMS.
I'M ALSO GOING TO WRITE THE LEFT SIDE AS X SQUARED + 0X,
WHICH IS STILL 0 + 8.
AND YOU'LL SEE WHY IN JUST A MOMENT.
SO WE'LL FACTOR OUT X SQUARED,
THAT WOULD GIVE US A + C x X SQUARED
+ AGAIN FACTORING OUT X WOULD GIVE US 4A + B x X + 4B.
AND NOW WE'LL EQUATE THE COEFFICIENTS;
MEANING THE X SQUARED TERMS ON THE LEFT AND RIGHT
MUST HAVE THE SAME COEFFICIENT.
THE X TERMS MUST HAVE THE COEFFICIENTS
ON THE LEFT AND RIGHT.
AND THE CONSTANTS ON THE LEFT AND RIGHT MUST BE THE SAME,
WHICH MEANS SINCE THE COEFFICIENT OF X
SQUARED IS EQUAL TO 1,
1 MUST BE EQUAL TO A + C,
0 MUST BE EQUAL TO 4A + B AND 8 MUST BE EQUAL TO 4B.
LET'S GO AND START WITH THE MOST BASIC EQUATION.
WE KNOW THAT 8 MUST EQUAL 4B WHILE 4B MUST EQUAL 8.
NEXT, WE KNOW THAT 4A + B MUST EQUAL 0
AND WE ALSO KNOW THAT A + C MUST EQUAL 1.
WELL, NOW WE'LL USE THIS SYSTEM OF EQUATIONS TO FIND THE VALUES
OF A, B, AND C,
WHICH WILL GIVE US OUR PARTIAL FRACTION DECOMPOSITION.
SO LOOKING AT THIS FIRST EQUATION;
IF 4B EQUALS 8 WE WOULD DIVIDE BOTH SIDES BY 4.
WE KNOW B EQUALS 2;
WELL IF B EQUALS 2 WE CAN USE A SECOND EQUATION
TO FIND THE VALUE OF A.
WE WOULD HAVE 4A + 2 EQUALS 0 SUBTRACT 2 ON BOTH SIDES,
WE HAVE 4A EQUALS -2 DIVIDE BY 4.
SO WE HAVE A = -1/2 AND THEN NOW THAT WE KNOW A = -1/2
WE CAN FIND C USING THE 3rd EQUATION.
WE WOULD HAVE -1/2 + C = 1, ADD 1/2 TO BOTH SIDES;
WE'D HAVE C = 1/2 OR C = 3/2.
SO NOW THAT WE HAVE THE VALUES OF A, B, AND C WE CAN GO
AND SUBSTITUTE THESE VALUES INTO OUR FRACTIONS HERE
FOR OUR PARTIAL FRACTION DECOMPOSITION.
LET'S GO AHEAD AND DO THIS ON THE NEXT SLIDE.
SO THE ORIGINAL RATIONAL EXPRESSION X SQUARED + 8
DIVIDED BY X TO THE 3rd + 4X SQUARED.
OF COURSE WE COULD LEAVE THE DENOMINATOR IN FACTORED FORM,
BUT SINCE A = -1/2 THIS IS THE SAME AS A x 1/X.
WE WOULD HAVE -1/2X.
NOTICE HOW THE 2 ENDS UP IN THE DENOMINATOR
WITH OUR FACTOR OF X + B DIVIDED BY X SQUARED, B IS 2.
SO 2 DIVIDED BY X SQUARED + C DIVIDED BY X + 4, BUT C IS 3/2.
X IS THE SAME AS C x 1/X + 4.
SO WE'D HAVE 3 DIVIDED BY 2 x THE QUANTITY X + 4.
NOW, NORMALLY YOU WON'T SEE THE FIRST TERM NEGATIVE
SO LET'S GO AHEAD AND CHANGE THE ORDER OF THIS.
LET'S GO AHEAD AND WRITE THIS AS 2 DIVIDED BY X SQUARED - 1/2X
+ 3 DIVIDED BY 2 x THE QUANTITY X + 4.
OBVIOUSLY, BOTH OF THESE FORMS ARE CORRECT,
BUT I THINK THE SECOND FORM IS PROBABLY MORE COMMON.
I HOPE YOU FOUND THIS EXPLANATION HELPFUL.