- WE WANT TO DIVIDE THIS DEGREE 3 POLYNOMIAL
BY THIS DEGREE 2 BINOMIAL,
AND SINCE A FRACTION BAR REPRESENTS DIVISION,
WE CAN WRITE THIS AS 2X CUBED - 8X SQUARED + 7X - 13
DIVIDED BY 2X SQUARED + 5.
NOTICE HOW IN THE DIVIDEND THE TERMS ARE IN DESCENDING ORDER,
MEANING FROM HIGHEST DEGREE TO LOWEST DEGREE,
AND ALL THE TERMS FROM DEGREE 3 TO DEGREE 0 ARE PRESENT.
SO TO START THIS DIVISION WE'RE ONLY CONCERNED
ABOUT THE FIRST TERM OF THE DIVISOR
AND THE FIRST TERM OF THE DIVIDEND.
AND THERE ARE TWO WAYS TO FIND
THE FIRST TERM OF OUR QUOTIENT OR OUR ANSWER.
ONE WAY IS TO ASK WHAT x 2X SQUARED = 2X TO THE 3rd?
IF WE KNOW THAT WOULD BE X,
THEN X WOULD BE THE FIRST TERM OF THE QUOTIENT.
OR, SINCE WE ARE PERFORMING DIVISION HERE,
WE CAN DIVIDE 2X CUBED BY 2X SQUARED,
AND THIS WOULD ALSO EQUAL X, OUR FIRST TERM.
NOW WE'RE GOING TO MULTIPLY X
AND OUR DIVISOR AND PUT IT BELOW.
SO WE'LL HAVE X x 2X SQUARED = 2X CUBED,
AND THEN WE HAVE X x 5 = 5X.
AND WE'RE GOING TO PUT THAT UNDERNEATH THE X TERMS
SO THAT LIKE TERMS ARE LINED UP, SO I'LL HAVE + 5X HERE.
BUT WHEN PERFORMING LONG DIVISION
WE NEED TO SUBTRACT THIS RESULT,
BUT INSTEAD OF SUBTRACTING THIS BINOMIAL,
WE'RE GOING TO ADD THE OPPOSITE,
MEANING WE CAN CHANGE THIS TO ADDITION
AS LONG AS WE CHANGE THE SIGN OF BOTH OF THESE TERMS.
SO WE CAN CHANGE THIS TO PLUS AS LONG AS WE CHANGE
THIS TO A NEGATIVE, AND THIS TO A NEGATIVE,
AND NOW WE'LL ADD.
2X CUBED + -2X CUBED = 0, AND THEN 7X + -5X = 2X.
AND THEN WE'LL BRING DOWN THE NEXT TERM,
WHICH IS GOING TO BE THIS TERM HERE.
SO WE'LL HAVE -8X SQUARED.
AND THIS IS POSITIVE, SO WE CAN WRITE THIS AS +.
AND NOW WE'RE ONLY CONCERNED
ABOUT THE FIRST TERM OF THE DIVISOR AND THIS FIRST TERM.
AND AGAIN, THERE'S TWO WAYS
TO FIND THE NEXT TERM OF OUR QUOTIENT.
WE CAN ASK WHAT x 2X SQUARED = -8X SQUARED?
IF WE KNOW THAT'S -4, WE CAN GO UP HERE AND WRITE - 4,
OR IF WE NEED TO, WE CAN DIVIDE -8X SQUARED BY 2X SQUARED,
WHICH IS ALSO EQUAL TO -4.
AND NOW WE'RE GOING TO MULTIPLY -4 WITH OUR DIVISOR.
SO -4 x 2X TO THE 2nd = -8X SQUARED, AND THEN -4 x 5 = -20.
SO WE HAVE - 20, AND AGAIN,
WE'RE GOING TO WRITE THIS UNDERNEATH
THE CONSTANT TERM IN OUR DIVIDEND,
SO WE'LL WRITE THAT OVER HERE.
AND NOW WE NEED TO SUBTRACT,
BUT INSTEAD WE'RE GOING TO ADD THE OPPOSITE.
SO AGAIN, WE'LL CHANGE THIS TO ADDITION,
AND THEN WE'LL CHANGE THE SIGN OF THIS TERM,
AND CHANGE THE SIGN OF THIS TERM.
AND NOW WE CAN JUST ADD VERTICALLY.
THIS WOULD BE 0.
BRING DOWN THE X TERM.
SO WE HAVE 2X, AND THEN WE HAVE -13 + 20 = 7.
NOTICE WE'VE USED ALL THE TERMS NOW AND THIS IS OUR REMAINDER.
NOTICE HOW THE DEGREE OF THE REMAINDER
IS LESS THAN THE DEGREE OF THE DIVISOR.
SO WE'LL HAVE A REMAINDER OF 2X + 7/DIVISOR 2X SQUARED + 5.
SO THIS IS A RESULT OF OUR DIVISION,
WHICH IS CALLED THE QUOTIENT.