- NOW WE'RE GOING TO TAKE A LOOK
AT FACTORING TRINOMIALS IN THE FORM AX SQUARED + BX + C
WHEN "A" IS NOT = TO 1 AND "A" IS NOT A COMMON FACTOR.
IN THIS VIDEO WE'LL USE THE GROUPING TECHNIQUE
TO FACTOR THESE TYPES OF PROBLEMS.
SO THE FIRST STEP IS TO RECOGNIZE
THAT 4 CANNOT BE FACTORED OUT OF THIS TRINOMIAL.
SO THE FIRST STEP IN THIS PROCESS
IS TO FIND THE FACTORS OF "A" x C THAT ADD TO B.
SO "A" x C WOULD BE 4 x -15 THAT'S -60.
WE WANT THE FACTORS OF-60 THAT ADD TO -4.
WELL, -6 x 10 IS = TO -60, BUT -6 + 10 IS = TO +4.
WE WANT A SUM OF -4.
SO IF WE SWITCH THESE SIGNS AND USE -10 AND +6
THAT'S = TO -60, AND THESE TWO FACTORS
DO HAVE A SUM OF -4.
SO -10, +6 ARE THE FACTORS THAT WE WANT TO USE.
SO NOW WHAT WE'RE GOING TO DO IS REWRITE -4X AS -10X + 6X.
SO OUR POLYNOMIAL WITH FOUR TERMS
WILL BE 4X SQUARED - 10X + 6X, AND THEN - 15.
AGAIN, NOTICE HOW WE HAVEN'T CHANGED THE TERM,
WE JUST BROKE IT UP INTO A SUM.
-4X IS = TO -10X + 6X.
IF YOU HAPPEN TO WRITE 6X - 10X, IT'S NOT GOING TO CHANGE
THE END RESULT.
BUT NOW THAT WE HAVE FOUR TERMS, WE'LL FACTOR THIS USING
THE GROUPING TECHNIQUE.
SO WE'LL DIVIDE THIS IN HALF, AND WE'LL FACTOR
THE LEFT AND RIGHT SIDE SEPARATELY,
HOPING FOR A COMMON BINOMIAL FACTOR.
SO THE GREATEST COMMON FACTOR BETWEEN 4X SQUARED AND 10X
WOULD BE 2X.
SO IF WE FACTOR OUT 2X WE'D BE LEFT WITH 2X - 5.
AND THE GREATEST COMMON FACTOR OF 6X AND 15 WOULD BE 3,
SO WE'LL FACTOR OUT A +3.
AND NOTICE HOW WE'RE LEFT WITH 2X - 5.
AND SO WE HAVE THIS COMMON BINOMIAL FACTOR HERE,
WHICH WILL NOW FACTOR OUT.
SO IF WE FACTOR OUT THE BINOMIAL FACTOR OF 2X - 5,
NOTICE HOW WE'RE LEFT WITH 2X + 3,
WHICH IS HOW WE FACTOR THE TRINOMIAL
4X SQUARED - 4X - 15 USING THE GROUPING TECHNIQUE.
SO YOU CAN SEE THIS PROCESS IS A LITTLE BIT LENGTHY,
BUT IF YOU LIKE FACTOR BY GROUPING, IT IS A NICE WAY
TO FACTOR A TRINOMIAL WHEN "A" DOESN'T EQUAL ONE.
LET'S TRY ANOTHER ONE. OKAY, AND FIRST STEP
IS TO FIND THE FACTORS OF "A" x C THAT HAVE A SUM OF B.
WELL, "A" x C WOULD BE 20 x 3.
SO WE WANT TO FIND THE FACTORS OF +60 THAT ADD TO B OR +19.
AND THAT WOULD BE 15 x 4.
THIS PRODUCT IS 60, AND 15 + 4 DOES = 19.
NOW WE'RE GOING TO REWRITE 19X AS 15X + 4X.
SO WE'D HAVE 20X SQUARED + 15X + 4X + 3.
NOW THAT WE HAVE FOUR TERMS, WE'LL DIVIDE THIS IN HALF
AND FACTOR BY GROUPING.
THE GREATEST COMMON FACTOR OF 20X SQUARED + 15X WOULD BE 5X
LEAVING US WITH 4X + 3.
NOTICE HOW 4X AND 3 DON'T HAVE ANY COMMON FACTORS,
BUT WE COULD FACTOR A 1 NOW.
SO WE'LL HAVE +1 X THE QUANTITY 4X + 3.
NOTICE HOW NOW WE HAVE A COMMON BINOMIAL FACTOR,
WHICH WILL NOW FACTOR OUT.
AND IF WE FACTOR OUT 4X + 3, NOTICE HOW WE'RE LEFT
WITH 5X + 1.
AND NOW OUR TRINOMIAL IS FACTORED
USING THE GROUPING TECHNIQUE.
AND WE'LL TAKE A LOOK AT ONE MORE METHOD
FOR FACTORING THESE TYPES OF TRINOMIALS IN THE NEXT VIDEO.