- WELCOME TO A VIDEO THAT WILL COVER THE BASICS
OF HOW TO FACTOR TRINOMIALS IN THE FORM OF AX SQUARED + BX + C,
WHEN A = 1, WHICH MEANS THE FIRST TERM WILL BE X SQUARED.
WE JUST FINISHED LEARNING HOW TO MULTIPLY TWO BINOMIALS TOGETHER,
MEANING WE WERE GIVEN SOMETHING IN FACTORED FORM
AND THEN WHEN WE MULTIPLIED IT OUT, IT WAS IN EXPANDED FORM.
AND NOW WE'RE GOING TO BE DOING THE OPPOSITE.
WE'RE GOING TO BE GIVEN A TRINOMIAL IN EXPANDED FORM
AND WE'RE GOING TO BE ASKED TO WRITE IT IN FACTORED FORM.
SO WE'LL BE GOING FROM THIS FORM TO THIS FORM.
LET'S TAKE A LOOK AT HOW WE'RE GOING TO DO THAT.
IF WE HAVE A TRINOMIAL AND THE LEADING COEFFICIENT IS 1,
MEANING IT'S IN THE FORM OF X SQUARED + BX + C,
IT'S A PRETTY STRAIGHTFORWARD PROCESS.
HERE'S WHAT WE'RE GOING TO DO.
WE'RE GOING TO MAKE TWO SETS OF PARENTHESES
AND PUT THE FACTORS OF X SQUARED IN THE FIRST POSITION
OF EACH SET OF PARENTHESIS.
WHEN THE LEADING COEFFICIENT IS X SQUARED,
THERE'S ONLY TWO FACTORS OF X SQUARED, AND THAT WOULD X AND X.
STEP TWO IS OUR MAIN TASK.
IN ORDER TO FIND THE TERMS THAT GO IN THE SECOND POSITIONS,
WE NEED TO FIND THE FACTORS OF C THAT ADD TO B.
BECAUSE THAT'S SO IMPORTANT, LET'S SAY THAT AGAIN.
WE NEED TO FIND THE FACTORS OF C THAT ADD TO B.
SO, FOR EXAMPLE, IF WE WANTED TO FACTOR X SQUARED + 8X + 12,
WE'D START BY MAKING TWO SETS OF PARENTHESES.
THE FIRST POSITIONS IN THE BINOMIALS
COME FROM THE FACTORS OF X SQUARED, THAT WOULD BE X AND X.
AND NOW WE NEED TO FIND THE FACTORS OF 12 THAT ADD TO 8.
NOW IT IS IMPORTANT THAT WE PAY ATTENTION TO THE SIGN.
NOTICE BOTH C AND B ARE POSITIVE.
SO, AGAIN, WE WANT THE FACTORS OF 12 THAT ADD TO 8.
NOW IF YOU KNOW YOUR MULTIPLICATION TABLES
REALLY WELL, YOU PROBABLY ALREADY CAN TELL
WHAT THOSE FACTORS WOULD BE.
IF YOU CAN'T, WHAT YOU MIGHT WANT TO DO
IS JUST LIST ALL OF THE FACTORS OF 12,
MEANING IT COULD BE 1 X 12, BUT THAT DOESN'T SUM TO 8.
2 X 6 AND 2 X 6 DOES ADD TO 8.
THEREFORE THESE ARE THE TERMS
THAT GO IN THE SECOND POSITION.
BOTH THESE TERMS ARE POSITIVE, SO WE'LL HAVE X + 2 AND X + 6.
NOW, WE COULD ALSO HAVE PUT THE 6 HERE AND THE 2 HERE.
REMEMBER, MULTIPLICATION IS COMMUTATIVE.
NOW, IF YOU DON'T KNOW YOUR MULTIPLICATION TABLES
ALL THAT WELL, YOU MIGHT FIND IT HELPFUL
TO HAVE A MULTIPLICATION TABLE OUT.
SO, FOR EXAMPLE, ON THE PREVIOUS PROBLEM,
IF I COULDN'T FIND THE FACTORS OF 12,
I COULD LOOK FOR 12s IN THIS TABLE.
FOR EXAMPLE, HERE'S A 12, WHICH TELLS ME TWO FACTORS OF 12
WOULD BE 3 AND 4.
THERE'S ANOTHER 12 HERE THAT TELLS ME
THE FACTORS OF 12 WOULD BE 6 AND 2.
AND THE ONLY ONE THAT'S MISSING FROM THIS TABLE WOULD BE 1 x 12,
BECAUSE THIS TABLE IS A 10 BY 10 MULTIPLICATION TABLE.
LET'S GO AND TAKE A LOOK AT SOME MORE EXAMPLES.
HERE WE HAVE X SQUARED - 4X - 32.
SO, SINCE OUR FIRST TERM IS X SQUARED,
WE CAN GO AHEAD AND MAKE TWO SETS OF PARENTHESES.
THE FIRST POSITIONS COME FROM THE FACTORS OF X SQUARED,
WHICH WOULD BE X AND X.
AND NOW OUR GOAL IS TO FIND THE FACTORS OF -32 THAT ADD TO -4.
AGAIN, NOTICE HOW WE DO HAVE TO PAY ATTENTION TO THE SIGN.
SO LET'S GO AHEAD AND LIST SOME OF THE FACTORS OF -32.
IT COULD BE -1 x 32 OR -32 X 1.
NEITHER OF THESE FACTORS ADD TO -4.
LET'S TRY -2 x 16 AND -16 x 2.
AGAIN, NEITHER OF THESE FACTORS ADD TO 4.
LET'S GO AHEAD AND TRY -4 x 8, OR MAYBE -8 x 4.
AND I THINK WE FOUND THE WINNING PAIR.
-8 x 4 DOES GIVE US -32.
AND THEN, WHEN WE ADD THESE FACTORS, -8 + 4 DOES GIVE US -4.
SO OUR TWO FACTORS ARE GOING TO BE X - 8 AND X + 4.
AGAIN, NOTICE THAT IF THE FACTOR IS NEGATIVE.
WE'RE GOING TO WRITE MINUS.
IF IT'S POSITIVE, WE'RE GOING TO WRITE PLUS.
OUR NEXT EXAMPLE, AGAIN, THE FACTORS OF Y SQUARED
WOULD BE Y AND Y.
HERE WE WANT THE FACTORS OF 36 THAT ADD TO -13.
ONE THING WE CAN NOTICE ON THIS PROBLEM IS,
IF WE WANT FACTORS OF 36 THAT ADD TO -13,
THE TWO FACTORS MUST BOTH BE NEGATIVE
TO GET A NEGATIVE SUM AND A POSITIVE PRODUCT.
SO WE'LL HAVE -1 x -36.
THAT DOESN'T ADD TO -13.
-2 x - 18, AGAIN, DOESN'T ADD TO -13.
HOW ABOUT -3 x -12?
WELL, THAT DOESN'T WORK EITHER.
BUT I THINK THE NEXT ONE WILL.
-4 X -9 DOES GIVE US 36 AND THESE FACTORS ALSO ADD TO -13.
SO WE'LL USE THESE AS THE TERMS OF THE SECOND POSITION.
SO ONE FACTOR WILL BE Y - 4.
THE OTHER FACTOR WILL BE Y - 9.
LET'S GO AND TAKE A LOOK AT A COUPLE MORE.
THE FACTORS OF X SQUARED WILL BE X AND X.
AND NOW WE WANT THE FACTORS OF - 12 THAT ADD TO A 1.
AND, AGAIN, IF YOU KNOW YOUR MULTIPLICATION TABLE,
YOU PROBABLY CAN ALREADY FIGURE OUT WHAT THAT WOULD BE.
BUT, IF NOT, WE'RE GOING TO GO AHEAD
AND LIST THEM OVER HERE ON THE RIGHT.
NOTICE, WE WANT THE PRODUCT TO BE NEGATIVE,
SO ONE FACTOR WILL BE POSITIVE AND ONE FACTOR WILL BE NEGATIVE.
SO, LET'S SEE, WE HAVE -1 x 12 OR -12 x 1.
THESE DO NOT HAVE A SUM OF 1.
-2 x 6 OR -6 x 2.
AGAIN, NEITHER OF THESE HAVE A SUM OF 1.
NEXT WOULD BE -4 x 3 OR -3 x 4.
AND ONE OF THESE WILL WORK.
IF WE WANT A SUM OF 1, WE'LL HAVE TO HAVE A 4 AND A 3.
SO OUR FACTORS WILL BE X - 3 AND X + 4.
AGAIN, WE CAN ALWAYS CHECK THESE BY MULTIPLYING IT OUT.
NEXT, WE HAVE X SQUARED + 28X + 75.
WE HAVE X AND X.
WE WANT THE FACTORS OF 75 THAT ADD TO A 28.
IN THIS CASE, WE'LL ONLY CONSIDER
THE POSITIVE FACTORS OF 75,
SINCE WE KNOW THE SUM HAS TO BE POSITIVE.
SO, OF COURSE, 1 x 75 WON'T WORK.
BUT 3 x 25 DOES GIVE US A 75 AND ALSO ADDS TO 28.
SO WE'LL HAVE X + 3 x X + 25.
AGAIN, WE COULD PUT THE 25 HERE AND THE 3 HERE.
IT WON'T CHANGE THE PRODUCT.
LET'S TAKE A LOOK AT TWO MORE.
NOW NOTICE ON THESE TWO PROBLEMS,
THE LEADING COEFFICIENT IS NOT JUST
X SQUARED OR JUST Y SQUARED.
NOW THE REASON I PUT THESE IN HERE
IS WE CAN'T FORGET THE FIRST STEP IN ANY FACTORING PROBLEM.
AND THAT FIRST STEP IS TO LOOK FOR THE GREATEST COMMON FACTOR.
AND ON THIS PROBLEM HERE,
THIS 3 IS A COMMON FACTOR AMONG ALL THREE TERMS.
SO WE CAN FACTOR OUT THE 3 FIRST AND SEE WHAT WE HAVE LEFT.
IF WE FACTOR OUT A 3 FROM X SQUARED, WE HAVE X SQUARED.
IF WE FACTOR OUT A 3 FROM 27X, THAT WOULD LEAVE 9X.
AND IF WE FACTOR A 3 OUT OF 42, WE'RE LEFT WITH 14.
NOTICE, NOW THAT WE'VE FACTORED OUT THE GREATEST COMMON FACTOR,
THE LEADING COEFFICIENT INSIDE THE PARENTHESES IS NOW A 1.
SO WE CAN PROCEED TO TRY TO FACTOR THE INNER TRINOMIAL.
LET'S GO AHEAD AND GIVE IT A TRY.
WE'D HAVE A FACTOR OF 3 AND THEN TWO SETS OF PARENTHESES.
THE FIRST POSITION WOULD CONSIST OF X AND X,
FROM THE FACTORS OF X SQUARED.
AND NOW WE WANT THE FACTORS OF 14 THAT ADD TO -9.
SINCE THE SUM HAS TO BE NEGATIVE,
WE'LL ONLY CONSIDER WHEN THE TWO FACTORS OF 14 ARE NEGATIVE.
SO -1 x -14, THAT DOESN'T SUM TO -9.
BUT -2 x -7 DOES GIVE US 14
AND, WHEN WE SUM THESE, WE DO GET -9.
SO WE'RE GOING TO HAVE X - 2 AND X - 7.
OKAY, IN OUR LAST EXAMPLE,
AGAIN THE LEADING COEFFICIENT IS NOT 1,
SO WE SHOULD LOOK FOR THE GREATEST COMMON FACTOR.
NOTICE ALL THESE TERMS HAVE A FACTOR OF 2
AS WELL AS A FACTOR OF Y IN COMMON.
SO WE'RE GOING TO FACTOR OUT 2Y.
WE'D BE LEFT WITH Y SQUARED - 6Y - 40.
AND, AGAIN, NOTICE THAT THE LEADING COEFFICIENT
INSIDE THE PARENTHESES IS 1.
SO WE CAN GO AHEAD AND FACTOR THE INNER TRINOMIAL
LIKE WE DID THE PREVIOUS EXAMPLES.
SO WE'LL HAVE 2Y AND THEN TWO SETS OF PARENTHESES.
WE'LL HAVE A Y HERE AND A Y HERE.
AND NOW WE WANT THE FACTORS OF - 40 THAT ADD TO -6.
SINCE THE PRODUCT MUST BE NEGATIVE,
WE'RE GOING TO HAVE ONE POSITIVE FACTOR AND ONE NEGATIVE FACTOR.
SO LET'S GO AHEAD AND START WITH -2 x 20 OR -20 x 2.
THOSE DON'T WORK BECAUSE THEY DON'T SUM TO -6.
LET'S GO AHEAD AND TRY -4 x 10 OR MAYBE -10 x 4.
NOW, I THINK WE'VE FOUND IT.
-10 x 4 DOES GIVE US -40 AND THE SUM IS ALSO -6.
SO WE'LL HAVE Y - 10 AND Y + 4.
THAT'LL DO IT FOR THE BASICS ON FACTORING A TRINOMIAL
WITH A LEADING COEFFICIENT OF 1.
THANK YOU FOR WATCHING.