- WELCOME TO THE BINOMIAL THEOREM USING PASCAL'S TRIANGLE.
THE GOAL OF THE VIDEO IS TO BE ABLE TO EXPAND A BINOMIAL.
LET'S START BY TAKING A LOOK AT THE EXPANSION OF X + Y.
WE NEED TO BE ABLE TO RECOGNIZE THE PATTERNS THAT ARE GENERATED
WHEN WE EXPAND THE BINOMIAL X + Y.
THE FIRST TWO ROWS SHOULD BE PRETTY OBVIOUS.
ANYTHING TO THE 0 POWER IS EQUAL TO 1
AND THEN ANYTHING RAISED TO THE 1st POWER
IS JUST EQUAL TO ITSELF.
HOWEVER, NOTICE WE ARE INCLUDING THE COEFFICIENTS HERE
AND I'LL SHOW YOU WHY IN JUST A MOMENT.
WHEN WE EXPAND THE QUANTITY X + Y SQUARED,
NOTICE THAT THE FIRST TERMS HAS 2 FACTORS OF X,
WHICH MATCHES THIS POWER HERE.
AND THEN THE NEXT TERM THE POWER ON THE X DECREASES TO 1,
BUT WE START PICKING UP FACTORS OF Y NOW.
AND IF WE SUM THESE 2 EXPONENTS
IT'S ALWAYS GOING TO EQUAL THE ORIGINAL EXPONENT
ON THE BINOMIAL.
AND THE THIRD TERM, WE HAVE 0 FACTORS OF X,
BUT WE HAVE 2 FACTORS OF Y.
AGAIN, NOTICE EACH OF THE EXPONENTS ALWAYS ADD TO 2,
WHICH IS THE SAME AS THE EXPONENT ON THE BINOMIAL.
IF WE TAKE A LOOK AT THE NEXT ROW,
WE HAVE THE QUANTITY X + Y TO THE 3rd POWER.
WE START OFF WITH 3 FACTORS OF X.
THE NEXT TERM HAS 2 FACTORS OF X AND 1 FACTOR OF Y,
AGAIN IT STILL ADDS TO 3.
THE NEXT TERM HAS 1 FACTOR OF X AND 2 FACTORS OF Y,
SO WE DROP A FACTOR OF X AND WE PICK UP A FACTOR OF Y
SO THE SUM OF THE EXPONENTS IS STILL 3.
AND THEN LASTLY, WE DROP ALL THE X VARIABLES
AND WE HAVE 3 FACTORS OF Y NOW.
AND THEN FINALLY THE LAST ROW, THE FIRST TERM X TO THE 4th,
NEXT TERM IS X TO THE 3rd, Y TO THE 1st.
THEN WE HAVE X SQUARED Y SQUARED,
X TO THE 1st, Y TO THE 3rd AND THEN FINALLY Y TO THE 4th.
NOW THESE COEFFICIENTS IN RED
ACTUALLY MAKE UP WHAT'S CALLED PASCAL'S TRIANGLE
AND THERE'S A SPECIAL WAY TO GENERATE THESE NUMBERS
BY STARTING WITH A TRIANGLE OF ONES AS WE SEE HERE.
AND THE WAY THAT'S DONE IS
EACH ROW STARTS WITH A ONE AND ENDS WITH A ONE
AND THEN THE TERMS IN THE MIDDLE ARE FORMED
BY ADDING THE TWO TERMS ABOVE IT.
1 + 1 = 2.
THE 3rd ROW, WE HAVE A 1 AND A 1 AT THE START AND THE END,
1 + 2 = 3, 2 + 1 = 3, AND THERE'S THE 1.
ON THE LAST ROW WE HAVE A 1 AT THE START AND 1 AT THE END.
1 + 3 = 4, 3 + 3 = 6, AND 3 + 1 = 4.
SO IF WE TAKE A LOOK AT JUST PASCAL'S TRIANGLE
IT WOULD LOOK SOMETHING LIKE THIS.
EVERY ROW STARTS AND ENDS WITH A ONE
AND THEN THE NUMBERS IN BETWEEN WE OBTAIN
BY ADDING THE TWO NUMBERS ABOVE IT.
SO FOR EXAMPLE, THE VERY LAST ROW
WE START WITH A 1, END WITH A 1.
1 + 5 = 6, 5 + 10 = 15, 10 + 10 = 20,
10 + 5 = 15, AND 1 + 5 = 6.
SO WHAT WE'RE GOING TO DO
IS COMBINE THE PATTERNS THAT WE SEE WITH THE VARIABLES
WITH PASCAL'S TRIANGLE THAT WE SEE HERE IN RED
TO EXPAND BINOMIALS.
AND ONE LAST THING TO MENTION IS THAT THE EXPONENT
ON THE ORIGINAL BINOMIAL
TELLS US WHICH ROW WE GO TO IN PASCAL'S TRIANGLE
TO FIND THE COEFFICIENTS
AS LONG AS WE COUNT THE FIRST ROW AS ROW ZERO.
HERE'S ROW ONE, TWO, THREE, FOUR.
SO ONCE WE KNOW THE EXPONENT
WE'LL KNOW WHICH ROW TO GO TO IN PASCAL'S TRIANGLE.
LET'S GO AHEAD AN SEE IF WE CAN APPLY THIS.
IF WE HAVE THE QUANTITY X + 3 RAISED TO THE 4th POWER,
LET'S FIRST DEAL WITH THE PATTERN OF THE EXPONENTS
IN EACH TERM
AND THEN WE'LL DEAL WITH THE MISSING COEFFICIENTS.
SO THE FIRST TERM IN THIS EXPANSION
WILL HAVE 4 FACTORS OF X AND 0 FACTORS OF 3.
THE NEXT TERM WILL HAVE 3 FACTORS OF X
AND WE'LL START PICKING UP FACTORS OF 3
SO WE HAVE 1 FACTOR OF 3.
AGAIN, NOTICE 3 + 1 DOES EQUAL 4.
PLUS THE NEXT TERM WILL HAVE 2 FACTORS OF X
AND 2 FACTORS OF 3,
THE NEXT TERM WILL HAVE 1 FACTOR OF X AND 3 FACTORS OF 3,
AND THEN FINALLY THE LAST TERM WILL HAVE 0 FACTORS OF X,
BUT 4 FACTORS OF 3.
AGAIN, NOTICE ALL ALONG HERE THE EXPONENTS ADD TO 4,
WHICH IS THE ORIGINAL EXPONENT ON THE BINOMIAL.
NOW WE DO HAVE TO GO TO PASCAL'S TRIANGLE,
GO TO ROW 4 TO WRITE IN THE COEFFICIENTS
OF EACH OF THESE PRODUCTS.
SO IF WE GO TO ROW 4 IN PASCAL'S TRIANGLE
WE'D HAVE 1, 4, 6, 4, AND 1.
LET'S GO AHEAD AND WRITE THOSE IN.
1, 4, 6, 4, AND 1.
NOW WE JUST HAVE TO SIMPLIFY EACH TERM.
SO WE WOULD HAVE X TO 4th
PLUS HERE WE'RE GOING TO HAVE A 4 x 3 = 12, X TO THE 3rd
PLUS HERE WE'RE GOING TO HAVE A 6 x 3 SQUARED = 9
SO 9 x 6 = 54X SQUARED
PLUS HERE WE HAVE A 4 x 3 TO THE 3rd.
WELL, 3 TO THE 3rd = 27 x 4 = 108X
AND THE LAST TERM WOULD BE 1 x 3 TO THE 4th = 81.
SO THIS IS A GREAT WAY TO EXPAND A BINOMIAL
BECAUSE THE ONLY OTHER WAY TO DO THIS
WOULD BE TO WRITE OUT 4 FACTORS OF X + 3
AND THEN JUST MULTIPLY IT 2 AT A TIME.
SO USUALLY THE BINOMIAL THEOREM USES COMBINATIONS
TO DETERMINE THESE COEFFICIENTS.
THAT WILL BE COVERED IN A LATER VIDEO,
BUT FOR RIGHT NOW PASCAL'S TRIANGLE WORKS JUST AS WELL.
I WANT TO SHOW THIS ONE
BECAUSE NOTICE HOW THIS ONE IS A DIFFERENCE.
SO THE FIRST TERM WILL BE 3X
AND WE'LL TREAT THE SECOND TERM AS A -2.
SO THE FIRST TERM WILL HAVE 3 FACTORS OF 3X,
THE NEXT TERM WILL HAVE 2 FACTORS OF 3X,
AND 1 FACTOR OF -2.
THE NEXT TERM WILL HAVE 1 FACTOR OF 3X AND TWO FACTORS OF -2,
AND THEN LASTLY WE'LL HAVE 0 FACTORS OF 3X
AND 3 FACTORS OF -2.
AGAIN, WE'LL GO TO ROW 3 IN PASCAL'S TRIANGLE.
HERE IN RED WE HAVE 1, 3, 3, 1.
AND NOW WE SIMPLIFY EACH TERM.
FIRST TERM WILL BE 3X CUBED,
THAT'S GOING TO BE 27X TO THE 3rd POWER.
THE NEXT TERM--FOR THE NEXT TERM WE'RE GOING TO HAVE 3
AND THIS WILL BE A 3 SQUARED,
SO WE'LL HAVE 3 x 9 = 27 x -2 = -54.
SO WE'LL WRITE -54 AND THEN WE HAVE 2 FACTORS OF X AS WELL.
THE NEXT TERM WILL HAVE 3 x 3 = 9, 9 x -2 SQUARED = 4,
9 x 4 = +36, AND THEN THE FACTOR OF X.
AND THEN LASTLY WE HAVE 1 x -2 TO THE 3rd POWER = -8,
SO WE'LL WRITE DOWN -8.
OKAY, AND THAT'S GOING TO PRETTY MUCH DO IT FOR THIS VIDEO.
WE WILL TAKE A LOOK AT THE BINOMIAL THEOREM AGAIN
AFTER WE COVER COMBINATIONS.
I HOPE YOU FOUND THIS HELPFUL.