- IN THIS VIDEO WE'LL 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 METHOD CALLED "BOTTOMS UP."
SO TO START THE BOTTOMS UP METHOD,
WE'RE GOING TO REWRITE THIS AS A TRINOMIAL WITH "A" = TO 1
BY REPLACING C WITH "A" x C.
SINCE "A" x C IS GOING TO BE 4 x -15,
WE'RE GOING TO REWRITE THIS AS X SQUARED - 4X - 60.
NOW, THESE TWO TRINOMIALS ARE NOT EQUIVALENT.
WE'RE GOING TO USE THIS TO FACTOR THE GIVEN TRINOMIAL.
SO WE'LL START BY FACTORING THIS AS WE NORMALLY WOULD.
AND SINCE "A" IS = TO 1,
WE'LL FACTOR THIS INTO TWO BINOMIALS
WHERE THE FIRST FACTORS WILL BE THE FACTORS OF X SQUARED,
WHICH ARE X AND X.
AND NOW WE WANT TO FIND THE FACTORS OF -60
THAT ADD TO -4.
WELL, -6 x 10 IS = TO -60, BUT -6 + 10 IS = TO +4.
WE WANT -4, SO IF WE SWITCH THE SIGNS OF THESE TWO FACTORS
AND USE -10 AND +6 IT'S A CORRECT PRODUCT
AND THE CORRECT SUM.
SO WE'LL HAVE X - 10 AND X + 6.
YOU'LL NEED TO BE CAREFUL HERE BECAUSE WE'RE NOT DONE.
NOW, SINCE WE MULTIPLIED -15 BY 4 TO GET THIS -60,
WE'RE GOING TO DIVIDE BOTH CONSTANT TERMS
OF THE BINOMIAL FACTORS BY 4.
NOW WE'RE GOING TO SIMPLIFY THESE TWO FRACTIONS.
HERE WE WOULD HAVE X - THIS WOULD BE 5/2,
AND HERE WE'D HAVE X + THIS WOULD 3/2.
HERE'S WHERE THE NAME "BOTTOMS UP" COMES FROM.
WE'LL TAKE THE DENOMINATORS HERE AND HERE,
THE NUMBERS ON THE BOTTOM, I'LL MOVE THEM UP
INTO THE POSITION OF THE COEFFICIENT
OF THE VARIABLE HERE AND HERE.
SO THE CORRECT FACTORING OF THE GIVEN TRINOMIAL
WOULD BE 2X - 5 x THIS WOULD BE 2X + 3.
AND OF COURSE IF WE WANTED TO, WE COULD FIND THIS PRODUCT
TO VERIFY THAT IT IS CORRECT.
LET'S TRY ANOTHER EXAMPLE OF THIS.
WE'RE GOING TO CREATE A TRINOMIAL WHERE "A" IS = TO 1
BY REPLACING C WITH A x C.
SO WE'LL HAVE X SQUARED + 19X, AND THEN 20 x 3 WOULD BE 60.
SO NOW THESE ARE NOT EQUAL, BUT WE'RE GOING TO USE THIS
TO FACTOR THE GIVEN TRINOMIAL.
AGAIN, WE REPLACED C WITH "A" x C,
AND MADE "A" = TO 1.
AND NOW WE'LL FACTOR THIS INTO TWO BINOMIAL FACTORS.
THE FACTORS OF X SQUARED WOULD BE X AND X.
NOW WE WANT THE FACTORS OF 60 THAT ADD TO 19.
THAT WOULD BE 15 x 4 SINCE 15 + 4 = 19.
SO WE'LL HAVE X + 15 AND X + 4.
AND NOW WE'RE GOING TO DIVIDE EACH CONSTANT,
OF THE BINOMIAL FACTORS, BY THE ORIGINAL "A",
OR IN THIS CASE 20.
NEXT STEP IS TO SIMPLIFY THESE FRACTIONS.
SO WE'LL HAVE X + THIS WILL BE 3/4.
AND HERE WE'D HAVE X + 1/5.
AND NOW WE'RE GOING TO MOVE THE SIMPLIFIED DENOMINATORS
HERE TO THE POSITION OF THE COEFFICIENTS OF X.
THIS IS WHY IT'S CALLED "BOTTOMS UP."
SO OUR TWO FACTORS ARE 4X + 3 AND 5X + 1.
THIS IS HOW YOU USE THE BOTTOMS UP METHOD
TO FACTOR TRINOMIALS WHEN "A" IS NOT = TO 1.