- WELCOME TO A SERIES OF VIDEOS ON FACTORING.
THE GOAL OF THIS VIDEO IS TO FACTOR A POLYNOMIAL
BY FACTORING OUT THE GREATEST COMMON FACTOR,
ALSO KNOWN AS THE GCF.
WE WILL NOW BEGIN TO LEARN METHODS
FOR FACTORING POLYNOMIALS.
FACTORING IS A TECHNIQUE THAT IS OFTEN USEFUL
WHEN TRYING TO SOLVE POLYNOMIAL EQUATIONS
ALGEBRAICALLY.
WE WILL START OUR STUDY OF FACTORING
BY LOOKING FOR THE GREATEST COMMON FACTOR OR GCF
OF A POLYNOMIAL EXPRESSION.
THE GCF IS THE LARGEST MONOMIAL THAT DIVIDES
OR IS A FACTOR OF EACH TERM OF THE POLYNOMIAL.
TO FACTOR AN EXPRESSION
MEANS TO WRITE AN EQUIVALENT FORM THAT IS A PRODUCT.
THE FIRST STEP IN FACTORING
IS ALWAYS TO LOOK FOR FACTORS COMMON IN EVERY TERM
TO IDENTIFY THE LARGEST OR GREATEST COMMON FACTOR.
THEN WE USE A DISTRIBUTIVE PROPERTY TO FACTOR OUT THE GCF
AND WRITE THE POLYNOMIAL AS A PRODUCT.
FOR EXAMPLE, IF WE WANTED TO FACTOR THE EXPRESSION 6X + 15,
WE COULD WRITE 6X AS 2 x 3 x X AND WE CAN WRITE 15 AS 3 x 5.
IF WE HAVE EACH TERM BROKEN DOWN INTO ITS PRIME FACTORS,
IT'S EASY TO IDENTIFY WHICH FACTORS THEY HAVE IN COMMON.
HERE WE SEE BOTH TERMS HAVE A FACTOR OF THREE
WITH NO OTHER FACTORS IN COMMON.
THEREFORE, THE GREATEST COMMON FACTOR IS THREE.
THEREFORE, WE CAN FACTOR THE THREE OUT
AND THE FACTORS NOT PART OF THE GCF
WOULD MAKE UP THE REMAINING EXPRESSION OF 2X + 5.
THIS IS CONSIDERED EXPANDED FORM
AND THIS IS CONSIDERED FACTORED FORM,
AND NOTICE THAT THIS IS EQUIVALENT TO 6X + 15,
BUT IT IS NOW WRITTEN AS A PRODUCT.
SO THE PROCESS TO FIND THE GCF OF A POLYNOMIAL WILL BE TO,
NUMBER ONE, WRITE EACH TERM IN PRIME FACTORED FORM.
STEP TWO, IDENTIFY THE FACTORS COMMON IN ALL TERMS
AND STEP THREE, FACTOR OUT THE GCF.
NOW, IF YOU KNOW YOUR MULTIPLICATION TABLES
REALLY WELL,
YOU MAY BE ABLE TO SKIP THIS FIRST STEP,
BUT I WILL SHOW IT FOR THESE EXAMPLES.
WE CAN REWRITE 2X TO THE FOURTH
AS 2 TIMES FOUR FACTORS OF X.
AND 6 IN PRIME FACTORED FORM WOULD BE FOUR FACTORS OF TWO
AND THREE FACTORS OF X
NOW WE CIRCLE THE FACTORS
THESE TWO TERMS HAVE IN COMMON.
THEY BOTH HAVE A FACTOR OF TWO
AND THEY BOTH HAVE THREE FACTORS OF X.
SO WHAT WE'VE CIRCLED IN EACH TERM
IS WHAT THEY HAVE IN COMMON,
WHICH GIVES US THE GREATEST COMMON FACTOR OF 2X CUBED.
SO WE CAN REWRITE THIS EXPRESSION IN FACTORED FORM
AS 2X CUBED TIMES X - 8.
IF WE WANT TO CHECK THIS, WE CAN JUST DISTRIBUTE AND SEE
THAT IT DOES MATCH THE ORIGINAL EXPRESSION.
OKAY, NOW THE REST OF THE PROBLEMS
WILL BE ESSENTIALLY THE SAME.
IT DOESN'T MATTER HOW MANY TERMS WE HAVE,
WRITE IT OUT IN PRIME FACTORED FORM,
IDENTIFY THE COMMON FACTORS AND THEN FACTOR OUT THE GCF.
SO FOR THIS NEXT EXAMPLE,
WE KNOW THAT FOUR IS EQUAL TO 2 x 2 TIMES TWO FACTORS OF X
AND THREE FACTORS OF 20 WOULD BE 4 x 5 AND FOUR IS 2 x 2.
SO WE HAVE 2 x 2 x 5,
ONE FACTOR OF X AND TWO FACTORS OF Y.
12 WOULD BE 4 x 3 AND FOUR AGAIN IS 2 x 2.
SO 2 x 2 x 3 x X x Y.
NOW, WE CIRCLE ALL OF THE FACTORS IN COMMON
AMONG ALL THREE TERMS.
THEY ALL HAVE A TWO AND ANOTHER TWO IN COMMON,
AND THEY ALL HAVE ONE FACTOR OF X AND ONE FACTOR OF Y.
SO WHATEVER WE'VE CIRCLED REPRESENTS THE GCF.
SO IN THIS CASE, WE'D HAVE 4XY AS OUR GCF.
AND AGAIN, WHATEVER WE HAVEN'T CIRCLED
IS WHAT'S LEFT IN OUR REMAINING EXPRESSION.
HERE WE HAVE XY SQUARED PLUS 5Y + 3.
AND AGAIN, WE COULD EASILY CHECK THIS IF WE WISH.
LET'S GO AHEAD AND TAKE A LOOK AT TWO MORE.
AND WHAT WE'LL NOTICE ABOUT THESE TWO EXPRESSIONS
IS THE LEADING COEFFICIENT IS NEGATIVE.
IN GENERAL, WHEN THE LEADING COEFFICIENT IS NEGATIVE,
WE FACTOR OUT A NEGATIVE GREATEST COMMON FACTOR.
SO LET'S GO AHEAD AND START BY WRITING THIS
IN PRIME FACTORED FORM.
SO WE SEE EACH HAS A FACTOR OF TWO
AND ALSO A FACTOR OF X
SO EVEN THOUGH THE GREATEST COMMON FACTOR IS 2X,
SINCE THE LEADING COEFFICIENT IS NEGATIVE,
WE WILL FACTOR OUT -2X.
BY FACTORING OUT A -2X,
THAT WILL CHANGE THE SIGN OF EACH TERM.
SO THE FIRST TERM WOULD BE NEGATIVE X SQUARED,
BUT SINCE WE'RE FACTORING OUT THE NEGATIVE,
IT WILL BE POSITIVE X SQUARED.
HERE WE HAVE 4X,
BUT INSTEAD OF PLUS 4X, IT WILL BE MINUS 4X.
AND HERE WE HAVE MINUS TWO,
BUT INSTEAD IT WILL BE PLUS TWO.
WE CAN CHECK OUR SIGNS BUT THEY WILL MATCH UP.
THIS IS THE PREFERRED FACTORED FORM
BY FACTORING OUT A NEGATIVE GREATEST COMMON FACTOR.
OKAY, ON THE LAST PROBLEM,
LET'S GO AHEAD AND WRITE IT OUT IN FACTORED FORM.
NOW, REMEMBER WE'RE LOOKING FOR TERMS
THAT ARE COMMON AMONG ALL OF THESE TERMS
AND THEY HAVE NOTHING IN COMMON
EXCEPT FOR ONE OR NEGATIVE ONE.
AND SINCE THE LEADING COEFFICIENT IS NEGATIVE,
WE WILL FACTOR OUT A NEGATIVE ONE,
WHICH WON'T CHANGE THE FACTORS OF THE TERMS,
JUST THE SIGNS.
SO WE'D HAVE A POSITIVE Y CUBED
PLUS 2Y SQUARED MINUS Y PLUS 7.
AND NORMALLY WE WILL NOT LEAVE THIS NEGATIVE ONE HERE.
WE'D JUST REWRITE THIS
AS NEGATIVE Y CUBED PLUS TWO Y SQUARED MINUS Y PLUS SEVEN.
I HOPE YOU FOUND THIS VIDEO HELPFUL
FOR IDENTIFYING AND FACTORING OUT THE GREATEST COMMON FACTOR
OF A POLYNOMIAL.
THANK YOU FOR WATCHING.