- WELCOME TO A VIDEO ON FACTORING
A PERFECT SQUARE TRINOMIAL.
TO FACTOR A PERFECT SQUARE TRINOMIAL,
WE NEED TO BE ABLE TO RECOGNIZE PERFECT SQUARES.
SO, IF WE HAVE A TRINOMIAL,
WHERE THE FIRST TERM IS A PERFECT SQUARE,
THE SECOND TERM IS A PERFECT SQUARE,
AND THE MIDDLE TERM IS TWICE A x B,
IT WILL BE A PERFECT SQUARE TRINOMIAL.
SO, IF WE TAKE A LOOK AT THIS FIRST FORMULA,
NOTICE THE FIRST POSITION OF THE TWO BINOMIAL FACTORS
IS EQUAL TO ONE FACTOR OF A SQUARED.
THE SECOND TERM OF THE TWO BINOMIAL FACTORS ARE EQUAL
TO ONE FACTOR OF THE THIRD TERM.
NOTICE THAT BOTH BINOMIALS ARE SUMS.
AND THIS COMES FROM THE FACT THAT THE MIDDLE TERM
IS POSITIVE WHEN WE SEE A PLUS SIGN HERE.
THE RESULT IS TWO EQUAL FACTORS,
AND THEREFORE, WE CAN WRITE A + B SQUARED.
AND EVERYTHING IS THE SAME IN THE SECOND FORMULA,
EXCEPT, NOW, THE MIDDLE TERM IS NEGATIVE WHERE WE HAVE -2AB,
AND THEREFORE, BOTH BINOMIALS ARE A DIFFERENCE.
AND, AGAIN, IF THE FIRST TERM IS "A" SQUARED,
WE'LL HAVE THE FACTOR OF "A" IN THE FIRST POSITIONS.
AND, IF THE THIRD TERM IS B SQUARED,
WE'LL HAVE A FACTOR OF B IN BOTH BINOMIALS.
SO WE HAVE THE QUANTITY "A" - B SQUARED.
NOW, THIS IS A LOT TO REMEMBER.
WHAT I SUGGEST IS, IF YOU NOTICE
THE FIRST TERM AND THE THIRD TERM ARE PERFECT SQUARES,
WELL GO AHEAD AND SET THIS UP AND JUST CHECK IT,
BECAUSE SOMETIMES IT'S TO RECOGNIZE THE MIDDLE TERM
BEING TWICE "A" x B.
NOW WE HAVE ENCOUNTERED THESE TYPES OF PROBLEMS
BEFORE WHEN WE LOOKED AT A LEADING COEFFICIENT OF ONE,
OR WHEN THE FIRST TERM WAS X SQUARED.
X SQUARED IS A PERFECT SQUARE.
SO, IF WE PUT ONE FACTOR OF X HERE AND ONE FACTOR OF X HERE,
AND THEN NOTICE THAT THE THIRD TERM IS ALSO A PERFECT SQUARE,
AND SINCE 5 SQUARED = 25,
WE COULD PUT ONE FACTOR OF 5 HERE AND ONE FACTOR OF 5 HERE.
AND THEN, LASTLY, SINCE WE HAVE -10X, WE HAVE TWO DIFFERENCES.
AND THE WAY WE CAN CHECK THIS
IS TO CHECK THE INNER PRODUCT AND OUTER PRODUCT
TO MAKE SURE WE HAVE A -10X.
AND WE DO.
WE HAVE -5X AS THE INNER PRODUCT,
-5X AS THE OUTER PRODUCT, WHICH DOES GIVE US THE -10X.
BUT, SINCE WE HAVE TWO EQUAL FACTORS,
WE CAN WRITE THIS AS THE QUANTITY X - 5 SQUARED.
NOW, I DO WANT TO MENTION,
WHILE THIS FORMULA IS A TIME SAVER,
THERE ARE SEVERAL OTHER METHODS TO FACTOR TRINOMIALS
WITH A LEADING COEFFICIENT NOT EQUAL TO 1.
AND I HAVE SEVERAL OTHER VIDEOS
THAT TALK ABOUT THE TRIAL-AND-ERROR METHOD,
THE GROUPING METHOD, AND A METHOD CALLED BOTTOMS UP.
SO, WHILE IF YOU DO RECOGNIZE THIS PATTERN,
IT WILL SAVE YOU SOME TIME,
THERE ARE OTHER WAYS TO FACTOR THESE TYPES OF PROBLEMS.
LET'S GO AND TAKE A LOOK AT SOME EXAMPLES.
AGAIN, NOTICE WE HAVE A TRINOMIAL.
THERE ARE NO COMMON FACTORS.
BUT WE CAN RECOGNIZE THAT THE FIRST TERM AND THE THIRD TERM
ARE PERFECT SQUARES.
SO, IF I RECOGNIZE THIS,
I WOULD JUST CHECK TO SEE
IF THIS IS A PERFECT SQUARE TRINOMIAL.
SO I WOULD CREATE MY TWO SETS OF PARENTHESES.
AND, SINCE 2X x 2X = 4X SQUARED,
I'D PUT A 2X HERE AND A 2X HERE.
AND BECAUSE 3 SQUARES EQUALS A 9,
I'D PUT A 3 HERE AND THE 3 HERE,
AND I'D MAKE THEM BOTH DIFFERENCES,
BECAUSE OF THE MINUS SIGN HERE IN THE MIDDLE.
AND NOW I WOULD CHECK IT TO MAKE SURE THAT IT DOES WORK,
MEANING THE INNER PRODUCT AND THE OUTER PRODUCT
SHOULD GIVE US A -12X.
IF IT DOES, THEN THIS IS A PERFECT SQUARE TRINOMIAL.
NOTICE THE INNER PRODUCT IS -6X,
THE OUTER PRODUCT IS ALSO -6X, WHICH DOES GIVE US -12X.
SO THIS IS A PERFECT SQUARE TRINOMIAL,
WHICH MEANS WE CAN WRITE THIS AS THE QUANTITY 2X - 3 SQUARED.
LET'S GO AHEAD AND TRY ANOTHER.
AGAIN, NOTICING THE FIRST TERM IS A PERFECT SQUARE
AS WELL AS THE THIRD TERM,
WE'RE GOING TO CHECK TO SEE
IF THIS IS A PERFECT SQUARE TRINOMIAL.
IF WE SQUARE 4X, WE'LL GET 16X SQUARED.
SO I'LL PUT A 4X HERE AND A 4X HERE.
AND, SINCE 7 SQUARED IS EQUAL TO 49,
WE'LL PUT A 7 HERE AND A 7 HERE.
AND THEN, LASTLY, SINCE WE HAVE A SUM IN THE MIDDLE,
WE'LL HAVE TWO SUMS IN OUR FACTORS.
NOW, LET'S GO AHEAD AND CHECK THIS.
THE INNER PRODUCT IS 28X, THE OUTER PRODUCT IS ALSO 28X,
WHICH DOES GIVE US 56X.
THIS DOES CHECK.
WE CAN REWRITE THIS AS THE QUANTITY 4X + 7
RAISED TO THE SECOND POWER.
LET'S GO AND TAKE A LOOK AT A COUPLE MORE.
HERE WE HAVE 25X SQUARED - 10X + 1.
AGAIN, NOTICE THE FIRST TERM IS A PERFECT SQUARE,
THE THIRD TERM IS A PERFECT SQUARE,
SO LET'S JUST GO AHEAD AND SEE IF WE CAN FACTOR THIS.
25X SQUARED IS THE SAME AS 5X RAISED TO THE SECOND,
SO WE'LL PUT A 5X HERE AND A 5X HERE.
AND, SINCE 1 SQUARED IS EQUAL TO 1,
WE'LL PUT A 1 HERE AND A 1 HERE.
AGAIN, WE HAVE A DIFFERENCE IN THE MIDDLE,
SO WE'LL TRY A - HERE AND A - HERE.
AND NOW WE'LL CHECK IT.
INNER PRODUCT IS -5X, OUTER PRODUCT IS -5X,
WHICH DOES GIVE US -10X, SO IT CHECKS.
IT IS A PERFECT SQUARE TRINOMIAL.
SO WE'LL HAVE THE QUANTITY 5X - 1 RAISED TO THE SECOND.
OKAY, ON THE LAST EXAMPLE,
BUT YOU HAVE TO BE A LITTLE MORE CAREFUL,
EVEN THOUGH THE FIRST TERM IS A PERFECT SQUARE
AND SO IS THE THIRD TERM,
WE CAN'T FORGET ABOUT THE FIRST STEP IN ANY FACTORING PROBLEM.
AND THE FIRST STEP IS TO LOOK FOR THE GREATEST COMMON FACTOR.
AND EACH OF THESE TERMS DO HAVE A COMMON FACTOR OF 4.
SO WE NEED TO FACTOR OUT THE GCF BEFORE WE DO ANYTHING ELSE.
IF WE FACTOR OUT 4, WE'LL HAVE X SQUARED + 4X + 4.
AND NOW WE CAN SEE IF WE CAN FACTOR THE TRINOMIAL.
SO WE'LL BRING DOWN OUR FACTOR OF 4
AND THEN LOOK FOR TWO BINOMIAL FACTORS.
NOW, HERE WE HAVE A TRINOMIAL WITH A LEADING COEFFICIENT OF 1.
SO WE CAN GO BACK TO THE OLD WAY OF FACTORING THIS,
WHERE WE REPLACE THE FACTORS OF X IN THE FIRST POSITION,
AND THEN DETERMINE THE FACTORS OF 4 THAT ADD TO 4.
THAT WOULD BE A 2 AND A 2.
SO THIS IS A PERFECT SQUARE TRINOMIAL.
BUT BECAUSE THE LEADING COEFFICIENT IS 1,
WE CAN USE A DIFFERENT TECHNIQUE, IF WE WISH.
SO THIS WOULD GIVE US 4 x THE QUANTITY X + 2
RAISED TO THE SECOND POWER.
NOW, I DO WANT TO SHOW THAT, IF WE DIDN'T FACTOR THE FOUR OUT,
WHAT IT WOULD LOOK LIKE.
WE MIGHT MAKE OUR TWO SETS OF PARENTHESES HERE.
AND THEN, SINCE THIS IS A PERFECT SQUARE,
WHERE 2X x 2X WOULD GIVE US 4X SQUARED.
BUT, SINCE 4 x 4 IS EQUAL TO 16,
WE MIGHT USE THESE FACTORS AND THEN USE A SUM.
THE PROBLEM WITH THIS IS THAT
NOTICE THERE IS A COMMON FACTOR OF 2 IN THIS BINOMIAL
AND ANOTHER COMMON FACTOR OF 2 IN THIS BINOMIAL.
AND, THEREFORE, IT'S NOT FACTORED COMPLETELY.
AND THAT'S WHY WE ALWAYS HAVE TO REMEMBER
THAT WE WANT TO FACTOR OUT THE GREATEST COMMON FACTOR FIRST,
BEFORE APPLYING ANY OTHER TECHNIQUE.
OKAY, I HOPE YOU FOUND THIS VIDEO HELPFUL.
THANK YOU FOR WATCHING.