- WELCOME TO ANOTHER VIDEO ON FACTORING TRINOMIALS,
WHEN THE LEADING COEFFICIENT, A, DOES NOT EQUAL 1.
THIS VIDEO SHOWS HOW TO FACTOR TRINOMIALS USING A METHOD
CALLED "BOTTOMS UP."
THE METHOD OF TRIAL AND ERROR AND GROUPING
ARE COVERED IN ANOTHER VIDEO.
SO THE MAIN PURPOSE OF BOTTOMS UP IS TO TRANSFORM A QUADRATIC
IN THIS FORM TO ONE WHERE THE LEADING COEFFICIENT = 1.
LET'S GO AHEAD AND TAKE A LOOK AT THE STEPS
IN AN EXAMPLE AT THE SAME TIME.
STEP ONE, WE MULTIPLY "A" x C
AND REWRITE IT IN THE FORM OF X2 + BX = AC.
SO FOR THIS PROBLEM WE WOULD REWRITE THIS
AS X SQUARED - 11X
AND THEN THE CONSTANT TERM WOULD BE 12 TIMES 2,
WHICH WOULD GIVE US 24.
THE NEXT STEP IS FACTOR AS WE NORMALLY WOULD
BY FINDING THE FACTORS OF THE CONSTANT TERM, AC,
THAT ADD TO B, THE COEFFICIENT OF THE X TERM.
SO FOR THIS PROBLEM WE WANT TO FIND THE FACTORS OF POSITIVE 24
THAT ADD TO -11-- THAT WOULD BE X - 3 x X - 8.
THE NEXT STEP IS WE DIVIDE THE CONSTANTS
IN EACH BINOMIAL FACTOR BY THE ORIGINAL VALUE OF "A."
THE ORIGINAL VALUE OF "A" WAS 12, SO WE DIVIDE THIS 3 BY 12,
AND WE DIVIDE THE 8 BY 12.
NEXT WE SIMPLIFY THE FRACTIONS THAT WE FORMED--
SO 3/12 WOULD SIMPLIFY TO 1/4, SO THIS WOULD BE X - 1/4,
AND 8/12 WOULD SIMPLIFY TO 2/3.
THE NEXT STEP, IF THE SIMPLIFIED FRACTION
DOES NOT HAVE A DENOMINATOR OF 1,
MOVE THE DENOMINATOR TO THE COEFFICIENT OF THE VARIABLE.
SO WHAT WE HAVE TO DO HERE IS TAKE THIS 4
AND MOVE IT TO THE COEFFICIENT OF THE X,
AND THE SAME THING WITH THIS 3--
WE MOVE IT TO THE COEFFICIENT OF THE X.
AND THIS WILL BE THE FACTORED FORM OF THE ORIGINAL POLYNOMIAL.
SO IN FACTORED FORM, WE WOULD HAVE 4X - 1 x 3X - 2.
AND YOU CAN SEE THE FIRST TERMS ARE THE FACTORS OF 12 X SQUARED
AND THE LAST TERMS ARE THE FACTORS OF 2.
AND OUR INNER PRODUCT AND OUTER PRODUCT DO GIVE US
THE MIDDLE TERM, SO IT CHECKS,
AND WE HAVE IT FACTORED CORRECTLY.
SO IT IS A VERY NICE TECHNIQUE TO REWRITE
WHAT CAN BE A CHALLENGING TRINOMIAL
INTO ONE THAT MOST OF US PREFER,
WHERE THE LEADING COEFFICIENT = 1.
LET'S TRY A COUPLE MORE OF THESE.
DON'T FORGET THAT WHENEVER WE FACTOR
WE SHOULD ALWAYS LOOK FOR A COMMON FACTOR FIRST.
THERE ISN'T ONE HERE, SO WE WILL MULTIPLY 8 x 15 x -6 = -90,
SO REWRITE THIS PROBLEM AS X SQUARED + X - 90.
AND WE FACTOR THIS AS WE NORMALLY DO--
WE WANT TO FIND THE FACTORS OF -90
THAT ADD TO POSITIVE THAT WOULD BE A POSITIVE 10
AND A -9--SO YOU'LL WRITE DOWN + 10 AND -9.
NOW, WE DON'T WANT TO STOP HERE--
REMEMBER THIS IS NOT THE ORIGINAL PROBLEM;
WE'RE DEALING WITH THIS PROBLEM HERE.
SO THE NEXT STEP IS WE DIVIDE THE CONSTANTS
BY THE ORIGINAL VALUE OF A, WHICH WAS 15,
AND THEN WE SIMPLIFY THESE FRACTIONS.
SO WE'D HAVE X PLUS, THIS WOULD BE 2/3,
AND THIS WOULD BE X - 3/5.
SO NOW WE TAKE THE DENOMINATORS OF THE CONSTANT TERMS
AND MOVE IT IN FRONT OF THE VARIABLE X
SO GOING BACK TO OUR ORIGINAL PROBLEM NOW--
THIS IS ALL OF OUR WORK--
IN FACTORED FORM THIS WOULD BE THE SAME AS 3X + 2
TIMES THE QUANTITY 5X - 3
AND YOU CAN GO AHEAD AND CHECK THAT BUT IT DOES CHECK.
LET'S GO AHEAD AND TRY ANOTHER.
NOW, AGAIN, IF WE GET IN A HURRY AND MULTIPLY 2 x 96,
WE'LL HAVE MADE AN ERROR BECAUSE 2 IS A COMMON FACTOR
AMONG THESE THREE TERMS.
SO WHEN WE FACTOR THE 2 OUT
WE'RE LEFT WITH X SQUARED - 14X + 48.
AND WE CAN SEE THAT THE INNER TRINOMIAL
HAS A LEADING COEFFICIENT OF 1,
SO FOR THIS PROBLEM WE DO NOT HAVE TO APPLY BOTTOMS UP;
WE CAN FACTOR THIS AS WE NORMALLY WOULD,
WHERE WE PUT THE FACTORS OF X SQUARED IN THE FIRST POSITIONS
AND TRY TO FIND THE FACTORS OF POSITIVE 48 THAT ADD TO -14.
AND WITH SOME TRIAL AND ERROR WE WOULD COME UP
WITH -8 x -6, WILL GIVE US 48, AND ALSO ADD TO -14.
SO YOU'VE GOT TO BE CAREFUL--
ALWAYS LOOK FOR YOUR GREATEST COMMON FACTORS
BEFORE YOU JUMP INTO ONE OF THE OTHER TECHNIQUES.
OKAY, LET'S TRY ANOTHER.
ON THIS PROBLEM WE DO HAVE A COMMON FACTOR OF 3,
SO LET'S FACTOR THE 3 OUT FIRST;
HOWEVER, THE INNER TRINOMIAL
STILL DOES NOT HAVE A LEADING COEFFICIENT OF 1.
SO WHEN WE FINISH WE KNOW WE SHOULD HAVE SOMETHING
THAT LOOKS LIKE THIS.
WHAT WE'RE GOING TO DO NOW IS TAKE THIS INNER TRINOMIAL
AND APPLY BOTTOMS UP.
SO LET'S REWRITE THIS AS X SQUARED - 5X
AND THEN THE CONSTANT TERM WHILE--AND THE CONSTANT TERM
WILL NOW BE 6 x -4, WHICH SHOULD GIVE US -24.
SO WE'LL FACTOR THIS--
SO WE'LL FACTOR THIS BY FINDING THE FACTORS OF -24
THAT ADD TO -5--THAT WOULD BE A -8, SO WE'LL WRITE "-8,"
AND THEN A 3, SO WE'LL WRITE IN "3."
BUT NOW WE HAVE TO REMEMBER
TO DIVIDE BY THE ORIGINAL VALUE OF "A".
WELL, THE ORIGINAL VALUE OF "A" WAS THIS 6 RIGHT HERE--
NOT THE 18 BUT THE 6.
SO WE DIVIDE EACH OF THESE BY 6, SIMPLIFY.
HERE WE'D HAVE X - 4/3, AND HERE WE'D HAVE X + 1/2.
HERE'S THE REASON WHY WE CALL IT "BOTTOMS UP"--
NOW WE TAKE ANY DENOMINATOR
AND MOVE IT UP TO THE POSITION OF THE COEFFICIENT.
SO THE RESULT WOULD BE 3 TIMES-- THIS WOULD BE 3X - 4,
AND THIS WOULD BE 2X + 1.
SO IT'S A VERY NICE METHOD AND MOST TEXTBOOKS DON'T SHOW THIS,
BUT FROM MY EXPERIENCE, ONCE STUDENTS KNOW THIS METHOD
THEY PREFER IT OVER THE OTHERS.
LET'S TAKE A MOMENT AND SEE WHY THIS ACTUALLY DOES WORK.
LET'S FIRST START BY SOLVING THIS
USING THE BOTTOMS-UP METHOD,
AND THEN WE'LL COMPARE IT TO ANOTHER TECHNIQUE
AND SEE WHY IT IS THE SAME.
SO IF WE WERE GOING TO USE BOTTOMS UP FOR THIS PROBLEM,
IT WOULD CHANGE THIS TO X SQUARED - 4X,
"A" x C WOULD BE A -12 OR - 12.
WE FACTOR THIS AS WE NORMALLY DO.
THE FACTORS OF -12 THAT ADD TO -4, THAT WOULD BE -6 + 2,
NOW DIVIDING BY THE ORIGINAL VALUE OF A, WHICH WAS 4--
AND IT'S IMPORTANT THAT WE DO NOT FORGET
WE HAVE TO SIMPLIFY NEXT--
SO WE WOULD HAVE X - 3 1/2, AND X + 1 1/2.
NOW WE PERFORM THE BOTTOMS UP,
SO IF THIS 2 GOES TO THE FRONT AND SO DOES THIS 2,
SO THE FINAL FACTORED FORM OF THE ORIGINAL
WOULD BE 2X - 3 x 2X + 1.
NOW, LET'S COMPARE THIS TO THE METHOD
WE USE WHEN THE LEADING COEFFICIENT IS EQUAL TO 1.
LET'S JUST SAY WE DECIDE TO FACTOR OUT 4.
WE'D BE LEFT WITH X SQUARED - X.
IF YOU FACTOR OUT 4 FROM 3 YOU'RE LEFT WITH 3/4.
WE CAN CHECK THIS, BUT THAT IS THE SAME.
NEXT WE WOULD BREAK IT UP INTO TWO BINOMIAL FACTORS--
THE FIRST POSITION WOULD BE THE FACTORS OF X SQUARED.
NOW, NEXT WE WANT TO FIND THE FACTORS OF -3/4 AND ADD TO -1.
NOW, THAT'S A PRETTY CHALLENGING QUESTION,
BUT THE FACTORS WOULD BE -3/2 AND POSITIVE 1/2.
NOW, WHAT I WANT YOU TO NOTICE, THIS STEP RIGHT HERE
AND THIS STEP RIGHT HERE ARE THE SAME
EXCEPT THIS HAS AN EXTRA FACTOR OF 4, AND THIS ONE DOESN'T.
BUT REALLY WHAT WE'RE DOING HERE
IS MULTIPLYING EACH OF THESE BINOMIALS BY 2.
TO SEE THAT, WHAT I COULD DO IS REWRITE THIS AS
"2 x X - 3 1/2 x 2 x X + 1 1/2."
AND WHAT WE'LL NOTICE, IF I DECIDED TO DISTRIBUTE THIS 2
INTO THE BINOMIAL FACTOR, WE WOULD HAVE 2X - 3;
AND THE SECOND BINOMIAL FACTOR WOULD BE 2X + 1.
SO WE MOVE THIS DENOMINATOR UP INTO THE COEFFICIENT OF X;
IT HAS THE SAME EFFECT AS MULTIPLYING BY 2,
AS YOU CAN SEE HERE, SO THE END RESULT IS THE SAME.
BUT THIS METHOD IS EASIER TO USE
BECAUSE OVER HERE ON THE RIGHT WE HAD TO DEAL WITH FRACTIONS.
NOW, THIS IS FAR FROM A PROOF OF THIS METHOD,
BUT YOU CAN KIND OF SEE FROM THIS EXAMPLE, ALGEBRAICALLY,
WHY THE RESULT IS EQUIVALENT.
I HOPE YOU FOUND THIS VIDEO HELPFUL.
THANK YOU AND HAVE A GOOD DAY.