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.