- WELCOME TO A VIDEO ON HOW TO FACTOR A DIFFERENCE OF SQUARES.
TO FACTOR A DIFFERENCE OF SQUARES,
WE NEED TO BE ABLE TO RECOGNIZE PERFECT SQUARE FACTORS.
SO, IF WE HAVE A BINOMIAL WHERE THE FIRST TERM
IS A PERFECT SQUARE,
THE SECOND TERM IS A PERFECT SQUARE,
AND WE HAVE A DIFFERENCE,
IT WILL FACTOR INTO THE QUANTITY A + B x THE QUANTITY A - B.
SO, FOR EXAMPLE, IF WE HAVE X SQUARED - 25,
X SQUARED IS A PERFECT SQUARE,
25 IS A PERFECT SQUARE
AND THEREFORE X SQUARED - 25 FACTORS INTO X + 5 x X - 5.
NOTICE, ONCE WE RECOGNIZE THE TERMS AS PERFECT SQUARES,
WE TAKE ONE OF THE FACTORS AND USE IT TO CREATE OUR BINOMIALS,
WHERE ONE IS A SUM AND ONE IS A DIFFERENCE.
AND, AGAIN, REMEMBER, SINCE MULTIPLICATION IS COMMUTATIVE,
MEANING YOU CAN CHANGE THE ORDER,
IT DOESN'T MATTER
WHETHER WE HAVE THE + 5 HERE OR THE - 5 HERE, AND VICE VERSA,
AS LONG AS ONE IS A SUM AND ONE IS A DIFFERENCE.
NOW THERE'S ONE MORE THING WE SHOULD NOTE.
IF WE HAVE A SUM OF SQUARES, IT WILL NOT FACTOR AND IS PRIME.
SO WE CAN ONLY FACTOR A DIFFERENCE OF SQUARES,
NOT A SUM OF SQUARES.
SO, IF WE WANT TO FACTOR X SQUARED - 81,
WE SHOULD RECOGNIZE THE FIRST TERMS OF PERFECT SQUARE,
THE SECOND TERMS OF PERFECT SQUARE,
AND WE HAVE A DIFFERENCE.
AND SINCE 9 SQUARED = 81,
ONE FACTOR WILL BE X + 9, THE OTHER FACTOR WILL BE X - 9.
NOW, JUST FOR AN ILLUSTRATION,
LET'S GO AHEAD AND MULTIPLY THIS OUT TO CHECK IT.
WE HAVE FOUR PRODUCTS, ONE, TWO, THREE, FOUR.
X x X, THAT WOULD BE X SQUARED.
AND THEN WE HAVE X x -9, - 9X.
OUR NEXT PRODUCT WILL BE 9 x X, THAT'S 9X.
AND THEN 9 x -9, WILL BE -81, SO WE'LL WRITE DOWN -81.
NOTICE THE MIDDLE TERMS ARE OPPOSITES,
AND SO THIS DOES CHECK.
WE HAVE X SQUARED.
THIS SHOWS WHY ONE HAS TO BE A SUM
AND ONE HAS TO BE A DIFFERENCE.
THIS RESULTS IN TWO TERMS BEING OPPOSITES,
GIVING US THE DIFFERENCE OF SQUARES.
NEXT, WE HAVE 64X SQUARED - 49.
AGAIN, THIS IS A PERFECT SQUARE, SO IS THIS.
AND WE HAVE A DIFFERENCE.
IF WE NEEDED TO, WE COULD SHOW SOME WORK HERE.
8X TO THE SECOND POWER DOES GIVE US 64X SQUARED.
AND 7 SQUARED, DOES GIVE US 49.
SO, IF THAT'S HELPFUL, WE CAN SHOW THIS.
WHAT THAT TELLS US IS: ONE FACTOR WILL BE 8X + 7.
THE OTHER FACTOR WILL BE 8X - 7.
AND, NOW, THIS DIFFERENCE OF SQUARES IS FACTORED.
LET'S GO AND TAKE A LOOK AT A COUPLE MORE EXAMPLES.
THESE NEXT TWO AREN'T QUITE AS STRAIGHTFORWARD.
HERE WE HAVE X TO THE FOURTH - 16.
WELL, X TO THE FOURTH IS A PERFECT SQUARE.
SO IS 16.
AND WE HAVE A DIFFERENCE.
NOW X TO THE FOURTH IS A PERFECT SQUARE BECAUSE X SQUARED,
RAISED TO THE SECOND POWER, DOES EQUAL X TO THE FOURTH.
AND, OF COURSE, 4 TO THE SECOND, DOES EQUAL 16.
SO OUR TWO FACTORS WOULD BE X SQUARED + 4 AND X SQUARED - 4.
NOW, WE HAVE TO BE CAREFUL HERE
BECAUSE THIS FACTOR HERE IS STILL A DIFFERENCE OF SQUARES.
NOW THIS ONE IS A SUM OF SQUARES,
SO IT CAN'T BE FACTORED.
BUT THIS CAN BE FACTORED AGAIN.
SO THIS FIRST FACTOR WOULD STAY THE SAME.
AND THIS IS GOING TO FACTOR INTO TWO MORE BINOMIALS.
X SQUARED IS A PERFECT SQUARE AND SO IS 4.
SO WE'D HAVE X + 2 AND X - 2.
REMEMBER, 2 SQUARED = 4.
SO THIS RESULTED IN THREE BINOMIAL FACTORS.
AND, IN OUR LAST EXAMPLE,
THIS FIRST TERM IS NOT A PERFECT SQUARE AND NEITHER IS 18.
BUT, REMEMBER, THE FIRST STEP IN ANY FACTORING PROBLEM
IS TO LOOK FOR THE GREATEST COMMON FACTOR,
AND 2 DOES HAPPEN TO BE A COMMON FACTOR BETWEEN THESE TWO TERMS.
IF WE FACTOR 2 OUT, WE'RE LEFT WITH X SQUARED - 9.
AND THIS BINOMIAL IS A DIFFERENCE OF SQUARES.
X SQUARED IS A PERFECT SQUARE, 9 IS A PERFECT SQUARE,
AND WE HAVE A DIFFERENCE.
SO WE CAN FACTOR THIS AGAIN.
WE BRING DOWN OUR FACTOR OF 2
AND THEN WE'D HAVE AN X + 3 AND AN X - 3.
AGAIN, REMEMBER, 3 SQUARED = 9,
AND THE FACTORS OF X CAME FROM THE FACTORS OF X SQUARED.
OKAY, THAT'S GOING TO DO IT FOR FACTORING DIFFERENCE OF SQUARES.
I HOPE YOU FOUND THIS HELPFUL.
THANK YOU FOR WATCHING.