- WELCOME TO ANOTHER VIDEO ON FACTORING TRINOMIALS.
THIS VIDEO WILL DISCUSS HOW TO FACTOR A TRINOMIAL
WHEN THE LEADING COEFFICIENT, "A", DOES NOT EQUAL 1.
WE'LL TALK ABOUT HOW TO FACTOR TRINOMIALS
USING THE TRIAL AND ERROR METHOD,
AND ALSO HOW TO USE THE GROUPING TECHNIQUE.
I WILL COVER HOW TO FACTOR A TRINOMIAL
USING A METHOD CALLED "BOTTOMS UP" IN ANOTHER VIDEO.
LET'S FIRST TALK ABOUT HOW TO USE THE TRIAL AND ERROR METHOD--
AND PLEASE KEEP IN MIND,
WE'LL ONLY USE THESE NEW METHODS
WHEN THE LEADING COEFFICIENT, "A", DOES NOT EQUAL 1,
OR WHEN IT CANNOT BE FACTORED OUT.
SO WHAT WE'LL DO IS WE'LL PLACE THE FACTORS
OF AX2 IN THE FIRST POSITIONS OF THE TWO SETS OF PARENTHESES
THAT REPRESENT THE FACTORS,
THEN WE'LL PLACE TWO POSSIBLE FACTORS OF C
INTO THE SECOND POSITIONS OF THE PARENTHESES.
THEN WE'LL FIND THE INNER AND OUTER PRODUCTS
OF THE TWO SETS OF PARENTHESES,
AND WE'LL KEEP DOING THIS UNTIL WE FIND
THE INNER AND OUTER PRODUCTS THAT ADD TO THE MIDDLE TERM, BX.
SO IF YOU KNOW
YOUR MULTIPLICATION TABLES REALLY WELL,
YOU MIGHT LIKE THIS METHOD;
IF YOU WANT A STEP-BY-STEP PROCESS THAT ALWAYS WORKS,
THIS MAY NOT BE THE BEST METHOD FOR YOU.
SO LET'S GO AHEAD AND GIVE THIS TRIAL AND ERROR METHOD A TRY.
IF WE WANT TO FACTOR THIS TRINOMIAL,
WE START OFF BY SETTING UP A SET OF PARENTHESES.
THE FIRST TERMS IN EACH FACTOR
MUST CONSIST OF THE FACTORS OF 6X SQUARED,
SO IT COULD BE 6X AND X OR 3X AND 2X.
I'M GOING TO GO AHEAD AND TRY THE 3X AND THE 2X.
NEXT WE LOOK AT THE FACTORS OF POSITIVE 15
AND PLACE THEM IN THE SECOND POSITIONS.
AGAIN, WE DON'T KNOW IF WE SHOULD USE 1 AND 15 OR 3 AND 5;
THAT'S WHY IT'S CALLED THE "TRIAL AND ERROR METHOD."
BUT THEY DO HAVE TO BE FACTORS OF POSITIVE 15.
SO LET'S SAY I DECIDE TO PUT THE 3 HERE AND THE 5 HERE--
SINCE I'M USING TWO POSITIVE FACTORS,
I'D HAVE TO USE PLUS SIGNS.
AND RIGHT AWAY I'M CONCERNED HERE
BECAUSE WE'RE NEVER GOING TO HAVE COMMON FACTORS
IN A BINOMIAL IF THERE WEREN'T ANY COMMON FACTORS
IN THE ORIGINAL TRINOMIAL.
BUT LET'S GO AHEAD AND TEST IT--
AND THE TEST IS CHECK THE INNER PRODUCT AND THE OUTER PRODUCT,
AND SEE IF THE SUM -- THE MIDDLE TERM OF -19X.
WE HAVE 21X.
SO THOSE ARE THE WRONG FACTORS.
SO A TRIAL AND ERROR METHOD MEANS THAT--
JUST TRY SOMETHING DIFFERENT.
SO LET'S GO AHEAD AND SET THIS UP AGAIN.
AND I STILL LIKE THE FACTORS OF 6X SQUARED AS 3X AND 2X,
BUT LET'S GO AHEAD AND SWITCH THE 5 AND THE 3 AROUND.
LET'S PUT THE 5 HERE THIS TIME
AND THE POSITIVE 3 HERE THIS TIME.
LET'S CHECK OUR INNER AND OUTER PRODUCT--
NOW WE HAVE A 10X PLUS 9X, WHICH EQUALS 19X.
NOW THIS IS ACTUALLY PRETTY GOOD.
WE HAVE A POSITIVE 19X, WE WANT A -19X.
BUT THAT'S ACTUALLY A PRETTY EASY FIX.
WE KNOW THAT 5 TIME 3 EQUALS 15, BUT SO DOES -5 TIMES -3,
SO LET'S TRY 3X MINUS 5 TIMES 2X MINUS 3.
LET'S CHECK OUR PRODUCTS.
NOW WE HAVE A NEGATIVE 10X AND A NEGATIVE 9X,
WHICH WOULD GIVE US THE NEGATIVE 19X THAT WE NEED.
SO THROUGH TRIAL AND ERROR
WE HAVE FOUND THE FACTORS OF THAT TRINOMIAL.
SO IF YOU DIDN'T FIND THIS PROCESS PLEASING,
YOU MAY PREFER THE GROUPING TECHNIQUE.
LET'S GO AHEAD AND TRY ONE MORE.
LET'S GO AHEAD AND SET UP OUR PARENTHESES,
WHICH REPRESENT OUR BINOMIAL FACTORS.
THE FIRST TERMS MUST CONSIST OF THE FACTORS OF 15X SQUARED,
SO LET'S TRY 5X AND 3X.
OF COURSE IT COULD BE 15X AND 1X, WE JUST DON'T KNOW.
NEXT, THE SECOND POSITIONS MUST COME FROM THE FACTORS OF -42.
SOME OF THE FACTORS OF -42 WOULD BE -7 TIMES 6, -6 TIMES 7,
-14 TIMES 3, -3 TIMES 14.
HOW DO WE KNOW WHICH FACTORS TO USE?
WELL, WE DON'T; WE JUST TRY TWO OF THEM--
THAT'S WHY IT'S CALLED "TRIAL AND ERROR."
HOWEVER, THE MORE YOU DO THIS THE BETTER YOU'LL GET AT IT.
SO IF WE TRIED, LET'S SAY, THE -7 HERE,
THEN THIS WOULD HAVE TO BE PLUS 6.
WELL, LET'S CHECK IT.
HERE WE HAVE 18X, AND THE OUTER PRODUCT WOULD BE -35X,
WHICH EQUALS -17X.
AGAIN, WE HAVE THE WRONG SIGN, BUT THE RIGHT NUMBER.
AND AGAIN, THAT'S A VERY EASY FIX--JUST SWITCH THE SIGNS.
INSTEAD OF USING THE POSITIVE 6, LET'S USE THE -6 HERE.
AND INSTEAD OF USING THE -7, WE'LL USE THE POSITIVE 7.
SO THIS WOULD ESSENTIALLY CHANGE THE SIGN OF THESE TWO TERMS,
WHICH WOULD CHANGE THE SIGN OF THE SUM,
AND WE HAVE FOUND OUR WINNER.
NOW, I'VE BEEN DOING THIS FOR QUITE A WHILE,
AND SO I'M--AND I HAVE A LITTLE MORE EXPERIENCE,
SO I CAN FIND THESE RATHER QUICKLY.
AND IF YOU CAN, TOO,
THIS'LL PROBABLY BE THE BEST METHOD FOR YOU.
BUT IF YOU WANT A DIFFERENT METHOD TO USE,
LET'S TRY THE GROUPING TECHNIQUE.
AND HERE'S WHAT WE'LL DO--
FIRST, WE NOTICE THAT A IS NOT A COMMON FACTOR,
SO WHAT WE DO IS WE FIND THE PRODUCT, AC,
THEN WE LIST THE FACTORS OF AC THAT ADD TO B,
THEN WE REWRITE BX AS A SUM
OR DIFFERENCE OF THE FACTORS OF AC THAT ADDED TO B,
AND THEN WE USE THE GROUPING TECHNIQUE.
NOW, IF YOU DON'T REMEMBER THE GROUPING TECHNIQUE,
YOU MAY WANT TO REVIEW THE VIDEO ENTITLED "FACTOR BY GROUPING."
OKAY, SO LET'S TRY THIS NEW METHOD.
STEP ONE, WE NEED TO FIND A TIMES C.
WELL, A TIMES C WOULD BE 9 TIMES 4, WHICH EQUALS 36.
SO NOW OUR QUESTION IS, WHAT ARE THE FACTORS OF 36
THAT ADD TO NEGATIVE 15?
LET'S LIST A FEW AND SEE IF WE CAN FIND IT.
-4 TIMES -9--THESE WOULD HAVE A SUM OF -13--
CLOSE, BUT NOT WHAT WE NEED.
NEXT WE MIGHT TRY -18 TIMES -2--
THAT WOULD BE -20-- THAT'S NOT WHAT WE WANT.
AND I THINK WE CAN FIND IT IF WE MULTIPLY -12 TIMES -3.
THIS MULTIPLIES THE POSITIVE 36 AND ADDS TO -15.
WHAT WE DO IS WE USE THESE TWO FACTORS
TO REWRITE THE MIDDLE TERM,
SO WHAT THAT MEANS IS WE WRITE "9X SQUARED..."
INSTEAD OF -15X WE'RE GOING TO WRITE "-12X"--"-3X"--
IT'S STILL -15X PLUS 4.
BUT NOW WE FACTOR THIS BY GROUPING,
SO WE'RE GOING TO CUT IT IN HALF,
FACTOR OUT THE GCF OF THE FIRST TWO TERMS--THAT WOULD BE 3X--
WE'RE LEFT WITH 3X MINUS 4.
NOW, THE SECOND TWO TERMS ONLY HAVE A COMMON FACTOR OF 1 OR -1.
REMEMBER, WE'RE LOOKING FOR ANOTHER BINOMIAL FACTOR
OF 3X MINUS 4.
IF WE FACTORED OUT A -1, WE'D HAVE A POSITIVE 3X,
AND THEN INSTEAD OF PLUS 4 WE'D HAVE A -4,
WHICH IS GOOD NEWS--WE HAVE A COMMON BINOMIAL FACTOR.
SO OUR LAST STEP, IF WE FACTOR OUT THE COMMON BINOMIAL FACTOR
OF 3X MINUS 4, WE'RE LEFT WITH 3X MINUS 1.
AND WE HAVE FACTORED THE ORIGINAL TRINOMIAL
USING THE GROUPING TECHNIQUE.
LET'S GO AHEAD AND TRY ANOTHER ONE.
FIRST, WE NOTICE THAT 10 IS NOT A COMMON FACTOR,
SO WE MULTIPLY A TIMES C--A TIMES C WOULD BE 10 TIMES -12,
WHICH WOULD EQUAL -120.
WE WANT TO FIND THE FACTORS OF -120 THAT ADD TO 7.
AGAIN, NOT A VERY EASY QUESTION--
ONE FACTOR IS GOING TO BE NEGATIVE, ONE WILL BE POSITIVE.
LET'S TRY-- LET'S JUST TRY SOME OF THESE.
-40 TIMES 3, THAT WOULD WORK,
BUT THAT'S OBVIOUSLY NOT GOING TO GIVE US
THE SUM OF POSITIVE 7.
IF WE TRIED -10 TIMES 12, THAT WOULD BE A SUM OF POSITIVE 2;
THAT WON'T WORK.
BUT IF WE TRIED -8 TIMES POSITIVE 15,
THAT WOULD GIVE US A SUM OF POSITIVE 7,
SO THOSE ARE THE TWO WINNING FACTORS.
ONE THING THAT I FIND HELPFUL
WHEN I'M TRYING TO FIND THESE FACTORS,
IS USE THE PRIME FACTORIZATION OF 120.
THIS IS WHAT I MEAN BY THAT-- IF I TAKE 120,
BREAK IT UP INTO 12 TIMES 10--
AND 12 WOULD BE 2 TIMES 2 TIMES 3, AND 10 WOULD BE 2 TIMES 5--
I CAN KIND OF PLAY AROUND WITH--
I CAN PLAY AROUND WITH THE PRIME FACTORIZATION
TO COME UP WITH THESE FACTORS.
BUT IT'S NOT AN EASY TASK.
OKAY, LET'S REWRITE THIS IN FOUR--
AS FOUR TERMS SO WE CAN USE GROUPING.
SO WE'LL HAVE 10 x 2--
AGAIN, INSTEAD OF 7X WE WILL USE THESE TWO FACTORS
TO WRITE IT AS -8X PLUS 15X MINUS 12,
CUT IT IN HALF AND USE GROUPING.
SO THE GCF OF THESE TWO TERMS WOULD BE 2X;
WE'D BE LEFT WITH 5X MINUS 4.
AND THEN HERE WE HAVE A COMMON FACTOR OF JUST 3,
SO WE'D HAVE 5X MINUS 4, WHICH IS GOOD NEWS
BECAUSE WE HAVE THAT COMMON BINOMIAL FACTOR AGAIN.
SO WE CAN COMPLETE THE FACTORING
BY FACTORING OUT THE 5X MINUS 4 AND WE'D BE LEFT WITH 2X PLUS 3.
SO THESE PROBLEMS ARE QUITE A BIT OF WORK,
BUT THEY ARE MANAGEABLE IF YOU TAKE IT STEP BY STEP.