- NOW WE'LL TAKE A LOOK AT FACTORING
PERFECT SQUARE TRINOMIALS.
THERE IS A SPECIAL FORMULA THAT WILL FACTOR
PERFECT SQUARE TRINOMIALS.
BUT LOOKING AT THESE FIRST TWO EXAMPLES,
MY SUGGESTION IS TO FACTOR THESE
AS WE NORMALLY WOULD WHEN "A" IS EQUAL TO ONE.
SO FOR X SQUARED + 12X + 36,
SINCE THERE ARE NO COMMON FACTORS,
IF THIS FACTOR'S JUST GOING TO FACTOR INTO TWO BINOMIALS,
THE FIRST POSITIONS OF THE BINOMIAL FACTORS
WILL COME FROM THE FACTORS OF X SQUARED,
WHICH WOULD BE X AND X.
AND NOW WE WANT THE FACTORS OF C
THAT ADD TO B OR THE FACTORS OF +36 THAT ADD TO +12.
NOTICE THAT 36 IS A PERFECT SQUARE
SINCE 6 x 6 IS = TO 36.
BUT ALSO NOTICE THAT 6 + 6 IS = TO 12.
SO THE TWO FACTORS WE NEED ARE + 6 AND +6.
SO WE HAVE X + 6 AND X + 6.
AND NOTICE THAT WE HAVE TWO EQUAL FACTORS,
WHICH MEANS WE CAN REWRITE THIS AS THE QUANTITY
X + 6 SQUARED, WHICH IS THE REASON
WHY THIS IS CALLED A PERFECT SQUARE TRINOMIAL.
AND LOOKING AT OUR SECOND EXAMPLE,
WE'LL APPLY THE SAME STRATEGY.
AGAIN, THERE ARE NO COMMON FACTORS HERE,
SO IF THIS DOES FACTOR, IT'LL FACTOR INTO TWO BINOMIALS.
THE FIRST POSITIONS OF THE FACTORS OF X SQUARED,
WHICH WOULD BE X AND X.
AND NOW WE WANT THE FACTORS OF +64 THAT ADD TO -16.
AGAIN, NOTICE THAT 64 IS A PERFECT SQUARE
BECAUSE 8 x 8 IS = TO 64.
BUT NOTICE THAT 8 + 8 IS = TO +16
AND WE WANT A SUM OF -16.
SO INSTEAD OF USING +8 AND +8, WE'LL USE -8 AND -8.
REMEMBER A NEGATIVE TIMES A NEGATIVE IS POSITIVE.
AND THEN -8 + -8 WOULD = -16, THE SUM THAT WE WANT.
SO OUR FACTORS ARE X - 8 AND X - 8.
AGAIN, THE TRINOMIAL FACTORS INTO TWO EQUAL FACTORS,
SO WE CAN WRITE THIS IN FACTORED FORM
AS THE QUANTITY X - 8 SQUARED.
THEREFORE, OUR TRINOMIAL IS A PERFECT SQUARE TRINOMIAL.
NOW, FOR THESE NEXT TWO EXAMPLES,
NOTICE HOW "A" IS NOT EQUAL TO 1.
BUT NOTICE IF IT IS A PERFECT SQUARE TRINOMIAL,
THE FIRST TERM AND THE THIRD TERM
MUST BE PERFECT SQUARES.
SO LOOKING AT OUR NEXT EXAMPLE, NOTICE HOW 4X SQUARED
IS A PERFECT SQUARE AND SO IS 9.
WE KNOW THIS BECAUSE 4X SQUARED WOULD BE EQUAL TO 2X SQUARED.
AND 9 IS A PERFECT SQUARE BECAUSE WE CAN WRITE THIS
AS 3 SQUARED.
SO NOW LET'S GO AHEAD AND FACTOR THIS
AS A PERFECT SQUARE TRINOMIAL AND THEN WE'LL CHECK
TO SEE IF IT WORKS.
SO WE'LL HAVE TWO BINOMIAL FACTORS.
THE FIRST POSITIONS MUST COME FROM THE FACTORS OF 4X SQUARED.
SO WE'LL USE 2X AND 2X.
THE SECOND POSITIONS MUST COME FROM THE FACTORS OF 9.
LET'S GO AHEAD AND TRY 3 AND 3.
AND THEN IF IT IS A PERFECT SQUARE TRINOMIAL,
WE'LL USE THE SIGN OF THE MIDDLE TERM
SO THESE WILL BOTH BE SUBTRACTION.
NOW, TO CHECK THIS, WE REALLY ONLY HAVE TO CHECK
TO MAKE SURE THE SUM OF THE INNER PRODUCT
AND THE OUTER PRODUCT DOES GIVE US THE MIDDLE TERM.
NOTICE THE INNER PRODUCT IS -6X.
THE OUTER PRODUCT IS -6X, WHICH WOULD HAVE A SUM OF -12X.
SO THIS IS A PERFECT SQUARE TRINOMIAL,
SO WE CAN REWRITE THIS AS THE QUANTITY 2X - 3 SQUARED.
NOTICE HOW IT DOES SAY HERE THE MIDDLE TERM
WOULD BE 2 x "A" x B.
THIS IS A LOT TO REMEMBER, AND SO I THINK
AS LONG AS WE CHECK TO MAKE SURE THE FIRST TERM
AND THE LAST TERM ARE PERFECT SQUARES
WE CAN QUICKLY CHECK TO MAKE SURE
THE INNER PRODUCT AND OUTER PRODUCT
HAVE A SUM OF THE MIDDLE TERM.
SO LOOKING AT ONE MORE EXAMPLE, AGAIN NOTICE
HOW "A" IS NOT = TO 1, THERE ARE NO COMMON FACTORS.
16X SQUARED IS A PERFECT SQUARE AND SO IS 25.
WE KNOW THIS BECAUSE WE CAN WRITE
16X SQUARED AS 4X SQUARED.
AND 25 IS A PERFECT SQUARE BECAUSE WE CAN WRITE THIS
AS 5 SQUARED.
SO NOW WE'LL FACTOR THIS INTO TWO BINOMIALS
AND SEE IF IT WORKS.
SO THE TWO FACTORS OF 16X SQUARE
THAT WE'RE GOING TO USE ARE 4X AND 4X.
THE FACTORS OF 25 WE'RE GOING TO USE ARE 5 AND 5.
WE USE THE SIGN OF THE MIDDLE TERM,
SO IT'S GOING TO BE PLUS HERE AND PLUS HERE.
AND NOW WE'LL JUST CHECK TO MAKE SURE THIS WORKS
BY CHECKING THE SUM OF THE INNER PRODUCT
AND OUTER PRODUCT.
WE HAVE 20X HERE AND WE HAVE 20X HERE,
WHICH DOES HAVE A SUM OF 40X, SO IT WORKS.
THEREFORE, WE HAVE A PERFECT SQUARE TRINOMIAL
AND WE CAN WRITE THIS AS THE QUANTITY
4X + 5 SQUARED.
AS I'VE SAID BEFORE, I'M NOT A BIG FAN
OF JUST MEMORIZING FORMULAS, BUT IN THIS CASE WE RECOGNIZE
THAT THE FIRST TERM AND THE THIRD TERM
ARE PERFECT SQUARES.
MY SUGGESTION IS TO GO AHEAD AND FACTOR IT
AS A PERFECT SQUARE TRINOMIAL AND THEN JUST CHECK
TO MAKE SURE IT WORKS.
I HOPE YOU FOUND THIS HELPFUL.