- WELCOME, THIS IS A FIRST OF A SERIES OF THREE VIDEOS
ON POLYNOMIAL DIVISION.
THIS VIDEO FOCUSES ON DIVIDING BY A MONOMIAL.
TO DIVIDE A POLYNOMIAL BY A MONOMIAL,
WE DIVIDE EACH TERM BY THE MONOMIAL.
IF WE LET "A," B, AND C REPRESENT REAL NUMBERS,
VARIABLES, OR ALGEBRAIC EXPRESSIONS,
SUCH THAT C DOESN'T = 0,
THE QUANTITY OF "A" + OR - B DIVIDED BY C = "A"
DIVIDED BY C + OR - B DIVIDED BY C.
NOW, THIS BECOMES PRETTY CLEAR
WHEN WE TAKE A LOOK AT A SPECIFIC EXAMPLE WITH VALUES
FOR "A," B, AND C.
SO FOR EXAMPLE, IF WE HAVE 1 + 2 OVER 3,
WE KNOW WE CAN ADD 1 + 2 TO GET 3, AND OF COURSE 3/3 = 1.
WELL, WHAT THIS IS ALSO TELLING US
IS WE COULD REWRITE THIS AS 1/3 + 2/3, AS WE SEE HERE.
WHICH WE KNOW WE CAN ADD,
BECAUSE WE HAVE A COMMON DENOMINATOR TO GET 3/3,
WHICH IS ALSO EQUAL TO 1.
SO THIS VERIFIES THAT IF WE ARE DIVIDING BY A MONOMIAL,
WE CAN BREAK IT UP INTO SEVERAL INDIVIDUAL FRACTIONS.
SO LET'S GO AHEAD AND GIVE IT A TRY.
OVER HERE ON THE RIGHT I HAVE PROVIDED THE QUOTIENT RULE
FOR EXPONENTS.
SO WE'RE DIVIDING BY THE MONOMIAL 4X,
WHICH MEANS WE CAN REWRITE THIS
AS 40X TO THE FOURTH / 4X - 24X SQUARED / 4X.
WELL, THIS SIMPLIFIES NICELY.
40 DIVIDED BY 4 = 10.
X TO THE FOURTH DIVIDED BY X TO THE FIRST,
REMEMBER WE SUBTRACT OUR EXPONENTS,
THAT WOULD GIVE US X TO THE THIRD.
24 DIVIDED BY 4 = 6.
AND X TO THE SECOND DIVIDED BY X TO THE FIRST = X TO THE FIRST,
OR JUST X.
NOW, WE CAN WRITE A DIVISION PROBLEM INVOLVING A MONOMIAL
IN A VARIETY OF WAYS.
AS YOU CAN SEE HERE,
IN NUMBER ONE IT WAS WRITTEN AS A FRACTION,
BUT WE CAN ALSO WRITE IT AS A DIVISION PROBLEM.
SO THIS WOULD BE THE SAME AS 32Y TO THE SIXTH
DIVIDED BY 4Y SQUARED - 8Y TO THE THIRD DIVIDED BY 4Y SQUARED,
AND LASTLY, + 2Y SQUARED DIVIDED BY 4Y SQUARED.
SO NOW WE JUST SIMPLIFY EACH FRACTION INDIVIDUALLY
TO OBTAIN OUR QUOTIENT.
36 DIVIDED BY 4 = 9.
Y TO THE SIXTH DIVIDED BY Y TO THE SECOND,
Y TO THE FOURTH - 8 DIVIDED BY 4 = 2.
Y TO THE THIRD DIVIDED BY Y TO THE SECOND,
3 - 2 WOULD GIVE US Y TO THE FIRST.
NOW HERE WE HAVE A FRACTION.
2/4 = 1/2, AND Y SQUARED / Y SQUARED = 1.
SO HERE IS OUR QUOTIENT.
PRETTY STRAIGHTFORWARD.
LET'S GO AHEAD AND TAKE A LOOK AT ONE MORE
THAT MAYBE LOOKS A LITTLE MORE INVOLVED,
BECAUSE WE DO HAVE A POLYNOMIAL WITH TWO DIFFERENT VARIABLES.
BUT AGAIN, THE PROCESS WILL BE THE SAME.
WE CAN SEE WE HAVE THREE TERMS IN THIS POLYNOMIAL.
THEREFORE, WHEN WE REWRITE THIS
WE'LL HAVE THREE SEPARATE FRACTIONS.
EACH FRACTION WILL HAVE A DENOMINATOR OF -3XY SQUARED.
SO NOW WE NEED TO SIMPLIFY EACH FRACTION INDIVIDUALLY.
SO THE FIRST STEP WE'LL LOOK AT THE COEFFICIENTS.
12 DIVIDED BY -3 = -4.
X TO THE THIRD DIVIDED BY X TO THE FIRST = X SQUARED.
Y TO THE SEVENTH DIVIDED BY Y TO THE SECOND = Y TO THE FIFTH.
9 DIVIDED BY -3 = -3.
BUT IT'S MINUS, SO - -3 = 3.
X TO THE SECOND DIVIDED BY X TO THE FIRST = X.
Y CUBED DIVIDED BY Y SQUARED = Y TO THE FIRST
+ 3 DIVIDED BY -3 = -1.
X DIVIDED BY X = 1,
AND Y SQUARED DIVIDED BY Y SQUARED = 1.
NOW, THIS LOOKS A LITTLE FUNNY HERE.
REMEMBER, ADDING A -1 IS THE SAME AS SUBTRACTING A +1.
SO LET'S GO AHEAD AND REWRITE THIS ONE MORE TIME AS - 1.
SO REMEMBER, THIS ONLY APPLIES
WHEN YOU'RE DIVIDING BY A MONOMIAL.
THE NEXT TWO VIDEOS WILL TAKE A LOOK
AT HOW YOU DIVIDE BY A BINOMIAL,
AND WE'LL TAKE A LOOK AT LONG DIVISION,
AND ALSO SYNTHETIC DIVISION.
THANK YOU FOR WATCHING, AND I HOPE YOU FOUND IT HELPFUL.