- WELCOME TO A VIDEO ON FACTORING TRINOMIALS
IN THE FORM AX2 + BX + C
WHERE A, THE LEADING COEFFICIENT, IS EQUAL TO 1.
WE WILL ALSO DISCUSS HOW TO FIND THE ZEROES
OF A QUADRATIC FUNCTION,
AND ALSO HOW TO SOLVE A QUADRATIC EQUATION
BY FACTORING.
LET'S TAKE A LOOK AT OUR STEPS TO FACTOR A QUADRATIC
IN THIS FORM.
STEP ONE, AS ALWAYS,
YOU WANT TO FACTOR OUT THE GCF IF PRESENT.
STEP TWO, WE'LL MAKE TWO SETS OF PARENTHESES
AND START BY PUTTING THE FACTORS OF X2
IN THE FIRST POSITION OF EACH SET OF PARENTHESES.
SINCE X TIMES X IS EQUAL TO X2,
OUR SETUP WILL LOOK LIKE THIS.
STEP THREE WILL LIST ALL OF THE FACTORS OF C,
THE CONSTANT TERM, PAYING CLOSE ATTENTION TO SIGN.
AND OUR MAIN GOAL IS TO DETERMINE THE FACTORS OF C
THAT ADD TO B,
THE COEFFICIENT OF THE X TERM.
ONCE WE FIND THOSE FACTORS
WE'LL PLACE THOSE IN THE SECOND POSITION
OF EACH SET OF PARENTHESES,
WHICH SHOULD RESULT IN THE TWO FACTORS OF THE QUADRATIC.
AND THEN, STEP FIVE,
WE CAN MULTIPLY TO CHECK THE FACTORED FORM.
SO LET'S GO AHEAD AND GIVE IT A TRY.
SO STEP ONE, WE'LL MAKE OUR TWO SETS OF PARENTHESES,
AND THE TWO FACTORS OF X2 WOULD BE X AND X.
THE NEXT STEP, WE'LL LIST ALL OF THE FACTORS OF -32--
SO WE HAVE -1 TIMES 32, -2 TIMES 16, -4 TIMES 8.
AND OF COURSE WE COULD ALSO HAVE -32 TIMES 1, -16 TIMES 2,
AND -8 TIMES 4.
ALL OF THESE MULTIPLIED TOGETHER GIVE US C,
WHICH IS -32.
WE WANT THE TWO FACTORS THAT ADD TO -14.
NOW, THESE TWO WOULD ADD TO POSITIVE 14,
SO RIGHT FACTORS, WRONG SIGNS.
SO THESE ARE THE TWO WINNERS RIGHT HERE.
THEREFORE, WE USE THESE FACTORS
TO WRITE THIS AS X MINUS 16 TIMES X PLUS 2.
SO IF IT'S A -16 SIXTEEN WRITE "-16,"
IF IT'S A POSITIVE 2 WE WRITE "+2."
AND OF COURSE WE COULD'VE PUT THE +2 HERE AND THE -16 HERE,
AND THE RESULT WOULD ESSENTIALLY BE THE SAME,
BECAUSE THE ORDER OF MULTIPLICATION
DOES NOT AFFECT THE PRODUCT.
FOR THIS FIRST ONE LET'S GO AHEAD AND CHECK IT--
WE'D HAVE X2 PLUS 2X MINUS 16 X MINUS 32--
SO EVERYTHING CHECKS OUT FINE HERE.
LET'S GO TO THE NEXT ONE.
NOW, IF YOU KNOW YOUR MULTIPLICATION TABLES
REALLY WELL
YOU COULD PROBABLY SKIP LISTING ALL THE FACTORS
OVER HERE ON THE RIGHT.
BUT FOR THE PURPOSE OF THIS VIDEO,
I AM GOING TO LIST THEM EVERY TIME.
SO WE HAVE THE FACTORS OF Y2-- Y TIMES Y.
NOW WE'LL LIST ALL OF THE FACTORS OF POSITIVE 36
ON THE RIGHT,
AND THERE'S QUITE A FEW OF THOSE--
2 TIMES 19, 3 TIMES 12, 4 TIMES 9.
I THINK WE HAVE THEM.
HOWEVER, WE COULD ALSO MULTIPLY
TWO NEGATIVES TOGETHER
AND STILL GET A POSITIVE 36.
AND THE REASON I'M STATING THAT
IS BECAUSE WE KNOW RIGHT AWAY
THAT WE WANT THE SUM OF THE FACTORS TO BE -15.
SO THESE TWO FACTORS GIVE US A SUM OF POSITIVE 15,
BUT AGAIN, WE WANT A SUM OF -15.
SO WHAT WE CAN DO IS WE CAN USE THIS INFORMATION
AND LIST IT AS -3 TIMES -12 INSTEAD.
SO SOMETIMES IT IS HELPFUL TO TAKE A GLANCE AT B
AND SEE WHAT WE WANT THE SIGN OF THE SUM TO BE.
THESE ARE THE WINNING FACTORS,
THEREFORE WE'RE GOING TO WRITE "Y MINUS 3 TIMES Y MINUS 12."
AND IF YOU WANT TO PAUSE IT AND CHECK THIS YOU CAN.
OKAY, ON THIS NEXT PROBLEM WE'LL SET IT UP
WITH OUR FACTORS OF X2.
WE WANT TO LIST THE FACTORS OF -12 THAT ADD TO A POSITIVE 1.
AS YOU GET BETTER AT THIS,
OR IF YOU KNOW YOUR MULTIPLICATION TABLES
REALLY WELL,
YOU CAN PROBABLY JUST DO THAT IN YOUR HEAD,
BUT I'LL GO AHEAD AND LIST THEM.
WE COULD HAVE POSITIVE 1 TIMES -12, OR -1 TIMES POSITIVE 12.
WE COULD HAVE POSITIVE 2 TIMES -6,
OR -2 TIMES POSITIVE 6, OR 3 TIMES -4, OR 4 TIMES -3.
AND THE TWO FACTORS THAT ADD TO POSITIVE 1
ARE POSITIVE 4 AND -3,
SO OUR FACTORS WILL BE X PLUS 4 TIMES X MINUS 3.
LET'S GO AHEAD AND TAKE A LOOK AT ANOTHER.
SO WE WANT THE FACTORS OF 75 THAT ADD TO A POSITIVE 20.
AS C GETS LARGER,
THIS QUESTION DOES BECOME MORE CHALLENGING.
WE COULD HAVE 25 TIMES 3.
I KNOW WE WANT TO HAVE A POSITIVE SUM,
AND I THINK WE JUST FOUND THE WINNING PAIR.
SO LET'S GO AHEAD AND STOP HERE
AND JUST WRITE IN "X PLUS 15 TIMES X PLUS 5."
LET'S GO AHEAD AND TAKE A LOOK AT TWO MORE.
NOW, ON NUMBER 5 THE FIRST THING WE NOTICE
IS THAT THE LEADING COEFFICIENT, A,
IS NOT EQUAL TO 1.
HOWEVER, THAT SHOULD NOT BE AN ISSUE
AS LONG AS THAT 3 CAN BE FACTORED OUT,
AND IN THIS CASE IT CAN.
SO IF WE FACTOR THE 3 OUT
WE'RE LEFT WITH X2 MINUS 9X PLUS 14.
AND SO SINCE THE EXPRESSION IN THE PARENTHESES
STILL HAS A LEADING COEFFICIENT OF ONE,
WE CAN USE THE SAME TECHNIQUE, EXCEPT WHEN WE SET THIS UP
WE'LL HAVE AN EXTRA FACTOR OF 3 IN THE FRONT.
WE STILL HAVE OUR FACTORS OF X2,
AND THEN WE WANT TO FIND THE FACTORS OF 14 THAT ADD TO -9.
WELL, AGAIN, INSTEAD OF LISTING ALL OF THEM,
WE REALLY HAVE TWO CHOICES--
WE WANT A POSITIVE 7 TIMES A POSITIVE 2,
OR A -7 TIMES A -2.
WHICH OF THESE WOULD GIVE US THE CORRECT SUM?
WELL, WE NEED TO HAVE A NEGATIVE SUM,
THEREFORE WE ARE GOING TO TAKE THE TWO NEGATIVE FACTORS,
THEREFORE THIS WOULD FACTOR
INTO 3 TIMES X MINUS 7 TIMES X MINUS 2.
THE SAME THING ON NUMBER 6--
THE GREATEST COMMON FACTOR OF THIS EXPRESSION IS "2Y."
WE FACTOR OUT 2Y, WE'RE LEFT WITH Y2 MINUS 6Y PLUS 40,
WHERE THE INNER QUADRATIC HAS A LEADING COEFFICIENT OF 1,
SO WE CAN APPLY THE SAME TECHNIQUE.
SO THE QUESTION BECOMES,
"WHAT ARE THE FACTORS OF POSITIVE 40 THAT ADD TO -6?"
LET'S LIST ALL THE FACTORS OF POSITIVE 40.
HOWEVER, ON THIS PROBLEM WE COULD SIT HERE ALL DAY--
NONE OF THESE FACTORS ADD TO - 6,
THEREFORE THIS PROBLEM DOES NOT FACTOR ANY FURTHER.
THE FINAL FACTORED FORM WOULD BE
2Y TIMES Y2 MINUS 6Y PLUS 40.
SO WE CAN ASSUME THESE WILL ALWAYS FACTOR,
AND HERE IS AN EXAMPLE OF THAT.
OKAY, LET'S ALSO MAKE THE CONNECTION
TO FINDING THE ZEROES OF A FUNCTION.
REMEMBER THAT THAT MEANS DETERMINE THE X VALUES
WHERE F OF X IS EQUAL TO 0.
GRAPHICALLY THAT WOULD BE
WHERE WE HAVE THE X INTERCEPTS.
SO LET'S GO AHEAD AND GET OUR GRAPHING CALCULATORS OUT
AND GRAPH THIS IN Y1.
LET'S PRESS "ZOOM 6"
TO MAKE SURE WE HAVE THE STANDARD WINDOW.
AND WE CAN SEE GRAPHICALLY--
I WON'T TAKE THE TIME TO CALCULATE IT
USING THE CALCULATE FEATURE--
HERE WE HAVE AN X INTERCEPT OF 7, ANOTHER ONE AT -8.
SO THOSE ARE THE ZEROES OF THE FUNCTION.
AND JUST TO MAKE THE CONNECTION
THAT IF WE WANTED TO WRITE THE FUNCTION IN FACTORED FORM,
IF POSITIVE 7 MADE THE FUNCTION EQUAL TO 0,
X MINUS 7 WOULD BE ONE OF THE FACTORS.
AND IF -8 IS A 0, X PLUS 8 WOULD NEED TO BE A FACTOR
SINCE WE'RE MULTIPLYING THESE TOGETHER
AND THE PRODUCT IS EQUAL TO ZERO.
USING THAT IDEA, WE CAN USE FACTORING
TO SOLVE QUADRATIC EQUATIONS USING THE 0 PRODUCT PROPERTY.
AND THIS IS WHAT I MEAN BY THAT.
FIRST, LET'S GO AHEAD AND FACTOR THIS.
AGAIN, WE HAVE A GCF HERE OF 5,
SO WE'D HAVE 5 TIMES X2 PLUS 2X PLUS 1
IS EQUAL TO FACTORING THIS WE WANT TO KNOW
WHAT ARE THE FACTORS OF POSITIVE 1
THAT ADD TO POSITIVE 2?
WELL, THAT WOULD BE POSITIVE 1 AND POSITIVE 1.
NOW, WE ACTUALLY WANT TO SOLVE THE EQUATION HERE,
NOT JUST FACTOR IT.
SO THE 0 PRODUCT PROPERTY SAYS,
"IF WE'RE MULTIPLYING TWO THINGS TOGETHER,
EITHER THE FIRST FACTOR OR THE SECOND FACTOR MUST EQUAL 0."
SO IF WE SOLVE THESE FOR X,
WE CAN SEE WE'LL HAVE X EQUALS -1 IN BOTH CASES,
THEREFORE "X EQUALS -1" IS THE SOLUTION.
BECAUSE IF WE SUBSTITUTE -1 IN HERE
WE'LL HAVE A 0 AND A 0 HERE.
WELL, 5 TIMES 0 WOULD GIVE US
LET'S JUST TRY ONE MORE OF THESE.
STEP ONE, DON'T FORGET, IF WE HAVE A GREATEST COMMON FACTOR
WE NEED TO FACTOR IT OUT.
AND THERE IS A COMMON FACTOR OF X HERE.
AND WE ACTUALLY CANNOT FACTOR THIS ANY FURTHER,
SO X TIMES X MINUS 9 MUST EQUAL ZERO.
SO EITHER X HAS TO EQUAL 0, OR X MINUS 9 MUST EQUAL 0.
WELL, THIS ONE'S ALREADY SOLVED FOR X,
SO THAT'S ONE SOLUTION.
AND THEN IF WE ADD 9 TO BOTH SIDES WE HAVE X EQUALS 9.
THESE ARE THE VALUES THAT MAKE THIS EQUATION EQUAL TO 0
NOW, WE'LL REVISIT THIS IDEA IN THE FUTURE,
BUT I DID WANT TO THROW IT IN NOW
JUST TO GET YOU THINKING ABOUT IT.
I HOPE YOU FOUND THIS VIDEO HELPFUL
AND THANK YOU FOR WATCHING.