- WELCOME TO A PRESENTATION ON LONG DIVISION WITH POLYNOMIALS.
THIS CAN BE A VERY LENGTHY PROCESS,
SO LET'S GO AHEAD AND GET STARTED.
IF THE DIVISOR IS NOT ONE TERM, OR A MONOMIAL,
WE USE LONG DIVISION TO DIVIDE POLYNOMIALS,
AND THIS IS PERFORMED VERY SIMILAR TO LONG DIVISION
IN ARITHMETIC IN THE UNITED STATES.
SO LET'S GO AHEAD AND REVIEW HOW WE WOULD PERFORM LONG DIVISION
ON THIS PROBLEM.
LET'S FIRST SET IT UP IN THE CORRECT FORM.
HERE WE HAVE THE DIVIDEND, AND HERE WE HAVE THE DIVISOR.
NOW, WE START BY LOOKING AT THE DIVISOR,
AND STARTING WITH THE LEFT DIGITS OF THE DIVIDEND,
HOW MANY 12s ARE THERE IN 25?
AND WE WOULD SAY 2, AND THEN WE'D MULTIPLY BY THE DIVISOR.
2 x 12 = 24.
AND THE NEXT STEP HERE IS WE SUBTRACT.
25 - 24 = 1.
WE BRING DOWN THE NEXT DIGIT,
AND WE CONTINUE THIS PROCESS UNTIL WE RUN OUT OF DIGITS.
SO HOW MANY 12s IN 14?
THAT WOULD BE 1.
1 x 12 = 12, AND WE SUBTRACT AND BRING DOWN THE 5.
THIS IS OUR LAST DIGIT, SO THIS IS OUR LAST CYCLE.
SO WE HAVE HOW MANY 12s IN 25?
THAT WOULD BE 2.
2 x 12 = 24 WITH A REMAINDER OF 1.
SO WE CAN REWRITE THIS AS 212 1/12.
WRITE THE REMAINDER OVER THE DIVISOR.
NOW, A DIVISION PROBLEM INVOLVING POLYNOMIALS
CAN TAKE A VARIETY OF FORMS.
WHICHEVER FORM IT'S WRITTEN IN,
WE DO HAVE TO WRITE IT IN LONG DIVISION FORM.
SO WE SET UP THE PROBLEM IN LONG DIVISION FORM
WITH THE DIVIDEND AND THE DIVISOR IN DESCENDING ORDER.
IF TERMS ARE MISSING, WE USE 0 TERMS.
WHAT THAT MEANS IS SINCE THIS POLYNOMIAL IS MISSING
THE DEGREE 2 TERM, WE HAVE TO INCLUDE A 0X SQUARED.
NOW WE ASK OURSELVES WHAT x THIS X
WOULD GIVE US X TO THE 3rd POWER?
AND THAT WOULD BE X SQUARED.
THAT'S THE FIRST TERM IN THE QUOTIENT.
NOW, WE COULD TAKE THIS TERM AND DIVIDE IT BY X.
THAT WOULD ALSO GIVE US X SQUARED.
AS SOON AS WE PUT A TERM UP HERE WE MULTIPLY IT BY THE DIVISOR.
SO WE'D HAVE X CUBED + 2X SQUARED.
AND REMEMBER WE WERE SUPPOSED TO SUBTRACT THIS,
BUT REMEMBER TO SUBTRACT POLYNOMIALS WE ADD THE OPPOSITE.
SO I CHANGE THIS TO ADDITION,
AS LONG AS I CHANGE THE SIGN OF THIS
AND CHANGE THE SIGN OF THIS,
THEREFORE THE RESULT WOULD BE -2X SQUARED.
BRING DOWN THE NEXT DIGIT, AND WE CONTINUE THIS PROCESS.
WHAT TIMES THIS X WOULD GIVE US -2X SQUARED?
AND THAT WOULD BE -2X.
NOW WE MULTIPLY THIS TERM BY THE DIVIDEND,
SO WE HAVE -2X SQUARED - 4X.
AND AGAIN, WE ARE SUPPOSED TO SUBTRACT,
BUT INSTEAD WE'LL ADD THE OPPOSITE.
SO IT CHANGES TO ADDITION.
CHANGE THAT SIGN, AND CHANGE THAT SIGN.
SO WE CAN SEE THIS WOULD GIVE US 0.
THIS WOULD BE -1X OR -X.
BRING DOWN THE LAST TERM, SO THIS IS THE LAST CYCLE.
AND WE ASK THAT QUESTION ONE MORE TIME,
WHAT TIMES X WOULD GIVE US -1X?
AND THAT WOULD BE -1.
MULTIPLYING BY THE DIVISOR, WE HAVE -X - 2,
BUT WE DO SUBTRACT THIS BY ADDING THE OPPOSITE.
SO CHANGE THE SIGN TO ADDITION, CHANGE THIS, AND CHANGE THIS.
SO WE CAN SEE OUR REMAINDER WILL BE 8.
SINCE IT'S 8 WHAT WE DO IS WE WRITE 8/X + 2,
AND THIS IS OUR FINAL QUOTIENT.
OKAY, LET'S TRY ANOTHER ONE.
STEP ONE, SET IT UP AS A LONG DIVISION PROBLEM.
THEN WE START BY ASKING OURSELVES WHAT TIMES
THE LEADING TERM OF THE DIVISOR
WOULD EQUAL THE LEADING TERM OF THE DIVIDEND,
OR WHAT x X = X SQUARED?
THAT WOULD BE X.
MULTIPLYING BY THE DIVISOR,
AND THEN SUBTRACTING BY ADDING THE OPPOSITE,
WE CAN SEE THE RESULT HERE.
THAT'D BE 0.
THIS WOULD BE -12X.
BRING DOWN THE NEXT TERM,
AND ASK THAT QUESTION ONE MORE TIME.
WHAT x X = -12X?
WELL, -12. MULTIPLYING BY THE DIVISOR, -12 x X,
-12 x 4, AND SUBTRACTING BY ADDING OPPOSITES.
SO THE RESULT HERE WOULD BE 28, SO WE WRITE THAT AS 28/X + 4.
HERE IS OUR QUOTIENT.
LET'S GO AHEAD AND TRY ANOTHER.
NOW HERE WE CAN SEE THE DIVIDEND IS MISSING SEVERAL TERMS,
SO WHEN WE SET THIS UP WE WILL HAVE SEVERAL 0 TERMS.
SO WE HAVE Y CUBED, SO WE MUST HAVE A Y SQUARED TERM,
SO THAT WOULD BE A 0 TERM.
THERE'S NO Y TERM, SO WE HAVE TO HAVE A 0Y TERM,
AND THEN + 27.
THEN WE START THE PROCESS OVER AGAIN.
WHAT x Y = Y CUBED?
WELL, THAT WOULD BE Y SQUARED.
MULTIPLYING BY THE DIVISOR,
AND SUBTRACTING BY ADDING THE OPPOSITE,
AND THE RESULT IS -3Y SQUARED.
BRING DOWN THE NEXT TIME, AND WE CONTINUE THE PROCESS.
WHAT x Y = -3Y SQUARED?
AND THAT WOULD BE -3Y.
MULTIPLYING BY THE DIVISOR, -3Y SQUARED - 9Y,
AND SUBTRACTING BY ADDING THE OPPOSITE.
SO CHANGE IT TO ADDITION, CHANGE THE SIGNS.
THE RESULT IS 9Y.
HERE COMES THE LAST TERM,
SO IT'S THE LAST CYCLE.
WHAT x Y = 9Y? 9.
MULTIPLYING BY THE DIVISOR,
AND SUBTRACTING BY ADDING THE OPPOSITES,
AND WE CAN SEE IN THIS CASE THE Y TERMS SIMPLIFY OUT
AND SO DO THE CONSTANTS.
SO OUR REMAINDER IS 0,
SO THERE IS OUR QUOTIENT.
AND WE CAN SAY THAT Y + 3 DIVIDES EVENLY
INTO Y CUBED + 27, SINCE WE HAVE A REMAINDER OF 0.
LET'S TAKE A LOOK AT ONE MORE.
LET'S SET IT UP.
SO IN THIS PROBLEM THERE WERE NO MISSING TERMS
IN DESCENDING ORDER, SO WE CAN GO AHEAD AND START.
WHAT x 8X = 24X CUBED?
WELL, 3 x 8 = 24, AND X x X SQUARED = X CUBED,
SO IT'S 3X SQUARED.
AGAIN, IF THAT'S DIFFICULT,
WE TAKE THE HIGHEST DEGREE TERM OF THE DIVIDEND
AND DIVIDE IT BY THE HIGHEST DEGREE TERM OF THE DIVISOR,
AND THIS WILL EQUAL 3X SQUARED.
MULTIPLYING BY THE DIVISOR NOW,
WE HAVE 24X CUBED + 9X SQUARED,
AND WE SUBTRACT BY ADDING THE OPPOSITE.
SO I'M GOING A LITTLE FASTER HERE,
BUT YOU CAN SEE THAT THE RESULT HERE WOULD BE -40X SQUARED.
BRING DOWN THE NEXT TERM.
WHAT x 8X = -40X SQUARED?
THAT WOULD BE -5 THAT WOULD GIVE US THE -40,
AND THEN X x X WOULD GIVE US X SQUARED.
MULTIPLYING BY THE DIVISOR, WE GET -40X SQUARED - 15X,
AND WE SUBTRACT HERE BY ADDING THE OPPOSITES.
IF YOU DO THIS CORRECTLY
THESE FIRST TERMS ALWAYS SIMPLIFY TO 0,
AND THE RESULT OF THIS LOOKS LIKE IT WOULD BE -56X.
AND WE'VE GOT ONE MORE TERM HERE.
THIS WILL BE OUR LAST CYCLE.
SO WHAT x 8X = -56X?
THAT WOULD BE -7.
NOTICE HOW THE TERMS IN THE QUOTIENT
ARE ALSO DECREASING BY 1 DEGREE EACH TIME.
THAT CAN BE HELPFUL TO MAKE SURE YOU HAVEN'T MADE A MISTAKE.
SO -7 x THE DIVISOR WOULD EQUAL -56X - 21.
BUT AGAIN, WE ARE SUBTRACTING BY ADDING THE OPPOSITES,
SO WE'LL CHANGE THIS, CHANGE THIS, AND CHANGE THIS.
LOOKS LIKE WE HAVE A REMAINDER OF -4,
SO WE WRITE OUR FINAL QUOTIENT AS THIS TRINOMIAL - 4/8X + 3.
AND NO, THE 4 AND 8 DO NOT SIMPLIFY.
THIS IS OUR FINAL QUOTIENT.
OKAY, I HOPE YOU FOUND THIS VIDEO HELPFUL.
THANK YOU AND HAVE A GOOD DAY.