- WELCOME TO A VIDEO ON MULTIPLYING POLYNOMIALS.
LET'S GO AHEAD AND GET STARTED.
LET'S START OFF BY TALKING ABOUT MONOMIALS.
TO MULTIPLY MONOMIALS, WE NEED TO MULTIPLY THE COEFFICIENTS
AND THEN MULTIPLY THE VARIABLES USING THE PRODUCT RULE
FOR EXPONENTS AS STATED HERE.
SO, IF WE WANT TO MULTIPLY 5X SQUARED x -3X TO THE 4th,
ALL OF THIS IS ATTACHED BY MULTIPLICATION,
INCLUDING 5X SQUARED, WHICH MEANS 5 x X SQUARED.
SO IF WE WANTED TO, WE COULD REARRANGE THE MULTIPLICATION
SO THAT THE COEFFICIENTS WERE NEXT TO EACH OTHER
AND THE VARIABLES WERE NEXT TO EACH OTHER.
REMEMBER, WE'RE ALLOWED TO CHANGE THE ORDER
OF MULTIPLICATION DUE TO THE COMMUTATIVE PROPERTY
OF MULTIPLICATION.
ONCE IT'S ARRANGED IN THIS ORDER,
IT'S A LITTLE BIT EASIER TO FIRST MULTIPLY
THE COEFFICIENTS TOGETHER,
5 x -3 = -15 AND X TO THE 2nd x X TO THE 4th
USING OUR PRODUCT RULE EQUALS X TO THE 6th.
MOVING ONTO NUMBER 2, IT MAY BE HELPFUL TO THINK OF A -1
AS THE COEFFICIENT TO THIS FIRST MONOMIAL.
SO AGAIN, IF WE REARRANGE THE ORDER
BY PUTTING THE COEFFICIENTS NEXT TO EACH OTHER,
-1 x -7 TIMES.
NOW, WE'LL ADDRESS THE Xs.
WE HAVE X, SO X TO THE 1st x X TO THE 2nd
x Y TO THE SECOND x Y TO THE 3rd.
NOW, WHEN WE MULTIPLY THE COEFFICIENTS,
WE'LL HAVE A 7, X TO THE 3rd, Y TO THE 5th.
NOTHING CHANGES WHEN WE HAVE THREE MONOMIALS
MULTIPLIED TOGETHER.
AND NOW, IF YOU DO THIS A WHILE,
YOU'LL PROBABLY SKIP THIS INTERMEDIATE STEP
AND FIRST MULTIPLY THE COEFFICIENTS TOGETHER,
4 x -2 WOULD BE -8 x 9 = -72.
"A" TO THE 3rd x "A" TO THE 1st x "A" TO THE 2nd
= "A" TO THE 6th.
OKAY, LET'S MOVE ALONG TO POLYNOMIALS.
TO MULTIPLY TWO POLYNOMIALS,
WE MULTIPLY EACH TERM IN ONE POLYNOMIAL BY EACH TERM
IN THE OTHER POLYNOMIAL AND THEN SIMPLIFY
BY COMBINING LIKE TERMS IN RIGHT AND DESCENDING ORDER.
SO, IF WE HAVE A MONOMIAL TIMES A BINOMIAL, WE HAVE TO MULTIPLY
THIS 3X TIMES THE 2X AND ALSO TIMES THE NEGATIVE 5.
NOW, I SAY NEGATIVE 5 BECAUSE I KNOW I CAN REWRITE THIS
IF I WANTED TO AS + -5 INSTEAD OF - 5.
SO, 3X x 2X OR 3 x 2 = 6,
X TO THE 1st x X TO THE 1st WOULD BE X TO THE 2nd
AND 3X x - 5 WOULD BE -15X,
BUT INSTEAD OF WRITING + -15X, WE'LL WRITE - 15X.
NUMBER TWO, WE NEED TO MULTIPLY -4X SQUARED
TIMES ALL THREE OF THE TERMS IN THE TRINOMIAL.
SO, THE FIRST PRODUCT WOULD BE NEGATIVE 12X TO THE 4th.
NEGATIVE 4X SQUARED x NEGATIVE 5X TO THE 1st
WOULD BE A 20X TO THE 3rd, SO PLUS 20X TO THE 3rd,
AND LASTLY, -4X SQUARED x 9 = -36X SQUARED.
NOW, LET'S TAKE A LOOK AT A BINOMIAL TIMES A BINOMIAL.
AGAIN, THE RULES STATES WE HAVE TO MULTIPLY EACH TERM IN THE 1st
TIMES EACH TERM IN THE 2nd.
SO, YOU COULD THINK OF THIS AS FIRST,
WE HAVE TO DISTRIBUTE THIS X INTO THE 2nd BINOMIAL
AND THEN, WE COULD DISTRIBUTE THE 3 INTO THE SECOND BINOMIAL.
SO, YOU CAN BE WE'RE FINDING THE PRODUCT OF EACH TERM IN THE 1st,
TIMES EACH TERM IN THE 2nd.
LET'S GO AHEAD AND TRY IT.
X TIMES X EQUALS X SQUARED.
I'LL MARK THAT OFF.
X TIMES 4 EQUALS 4X, 3 TIMES X EQUALS 3X,
AND 3 TIMES 4 WOULD BE 12.
NOW, YOU CAN SEE HERE,
WE DO HAVE TWO LIKE TERMS IN THE MIDDLE,
SO OUR FINAL SIMPLIFIED PRODUCT WOULD BE X SQUARED + 7X + 12.
NOW, THERE IS A GEOMETRIC WAY TO REPRESENT THIS PRODUCT.
FOR EXAMPLE, IF WE CALLED THE LENGTH OF THIS SIDE X + 3,
WE COULD CALL THIS YELLOW LENGTH X AND THIS BLUE LENGTH 3
AND IF THIS SIDE HAD LENGTH X + 4,
WE COULD LABEL THIS LENGTH X AND THIS LENGTH 4.
AND OF COURSE THE AREA OF THIS RECTANGLE
WOULD BE LENGTH TIMES WIDTH.
SO, IF WE TAKE A LOOK AT THIS,
THE AREA OF THE YELLOW REGION WOULD BE X TIMES X OR X SQUARED.
THE AREA OF THIS BLUE REGION HERE
WOULD BE 3 TIMES THIS LENGTH, WHICH IS X, SO THIS WOULD BE 3X.
THE OTHER BLUE REGION WOULD BE 4 TIMES X OR 4X
AND THIS LAST REGION HERE
WOULD BE 3 TIMES 4 = 12.
SO AGAIN, TO FIND THIS TOTAL AREA
BY FINDING THIS PRODUCT WOULD RESULT IN THE SAME PRODUCT
WE FOUND HERE ON THE LEFT.
ANOTHER WAY TO THINK ABOUT IT, TO MAKE SURE THIS IS LOGICAL,
LET'S ASSUME THAT X = 4,
WELL IF X = 4, WE'D HAVE 4 + 4 OR 8 AND OVER HERE,
WE HAVE 4 + 3 OR 7
WE KNOW THE AREA OF A RECTANGLE
THAT'S 7 BY 8 = 56.
LET'S JUST VERIFY THAT THIS PRODUCT
IS ALSO EQUAL TO 56 BY SUBBING 4 IN FOR X.
SO, WE HAVE 4 SQUARED + 7 x 4 = 12, 16 + 28 + 12;
THIS DOES EQUAL 56.
LET'S GO AHEAD AND TAKE A LOOK AT A COUPLE OF MORE EXAMPLES.
HERE, WE HAVE ANOTHER BINOMIAL TIMES A BINOMIAL.
SO, WE'LL START WITH THIS X AND MULTIPLY IT BY BOTH TERMS
IN THE OTHER BINOMIAL.
SO, WE'LL HAVE X x X,
X x -2 AND WE'LL MULTIPLY THE 5 TIMES BOTH TERMS
IN THE SECOND BINOMIAL.
AGAIN, YOU CAN SEE WE HAVE 4 PRODUCTS.
THE FIRST PRODUCT, X TIMES X EQUALS X SQUARED,
X TIMES A NEGATIVE 2,
NEGATIVE 2X OR MINUS 2X,
5 TIMES X EQUALS 5X,
AND POSITIVE 5 TIMES NEGATIVE 2 EQUALS NEGATIVE 10.
AGAIN, INSTEAD OF WRITING PLUS OR NEGATIVE 10,
I'LL WRITE MINUS 10, WHICH MEANS THE SAME THING.
NOW, WE COMBINE OUR LIKE TERMS.
SO, OUR FINAL PRODUCT X SQUARED + 3X - 10.
AND NUMBER TWO, IT'S THE SAME PROCEDURE.
WE MULTIPLY THIS 4X SQUARED TIMES BOTH TERMS IN THE SECOND
AND THEN MULTIPLY NEGATIVE 1 TIMES BOTH TERMS IN THE SECOND.
SO, LET'S TAKE IT ONE STEP AT A TIME.
WE MAY WANT TO WRITE THIS AS 2X TO THE 1st TO HELP US
OR X SQUARED TIMES 2X WOULD BE 8X CUBED
OR X SQUARED TIMES -3 = 12X SQUARED -1 x 2X - 2X,
AND -1 x -3 WOULD EQUAL 3.
WE CAN SEE, WE DO NOT HAVE ANY LIKE TERMS
AND IT'S ALREADY IN DESCENDING ORDER,
SO WE'RE FINISHED HERE.
IN THE LAST EXAMPLE,
WE HAVE A BINOMIAL TIMES A TRINOMIAL
AND AGAIN, THE PROCESS STAYS THE SAME,
THEN MULTIPLY THIS X TIMES EACH OF THESE TERMS IN THE 2nd.
AGAIN, YOU CAN THINK OF DISTRIBUTING THE X
INTO THE 2nd TRINOMIAL.
THEN, WE HAVE TO MULTIPLY THIS -4 x 3 TERMS IN THE 2nd.
SO, IT'S ALMOST LIKE A DOUBLE DISTRIBUTION.
AND YOU CAN SEE WE'LL HAVE 6 PRODUCTS.
SO, THE FIRST PRODUCT, X TO THE 1st x X TO THE 2nd,
X TO THE 3rd, X x 5X + 5X SQUARED, X x -7.
NOW, WHEN I WRITE THE NEXT THREE PRODUCTS,
I'M GOING TO WRITE THEM IN VERTICAL FORM.
WHAT I MEAN BY THAT IS HERE I HAVE NEGATIVE 4 TIMES X SQUARED.
THAT'S NEGATIVE 4X SQUARED.
I'LL PUT THAT UNDERNEATH THE X SQUARED TERM.
THIS WILL FACILITATE COMBINING LIKE TERMS.
NEXT, WE HAVE -4 x 5X OR -20X OR - 20X AND -4 x -7
WOULD BE A 28.
LAST STEP TO COMBINE OUR LIKE TERMS,
WE WOULD HAVE X CUBED + 1X SQUARED - 27X + 28.
I HOPE YOU FOUND THIS VIDEO HELPFUL.
THANK YOU FOR WATCHING AND HAVE A GOOD DAY.