- NOW WE'LL LOOK AT SOME EXAMPLES OF FACTORING TRINOMIALS IN THE FORM AX SQUARED + BX + C WHEN "A" IS NOT = TO 1 AND "A" IS NOT A COMMON FACTOR. HERE WE HAVE 4X SQUARED - 4X - 15. NOTICE THAT 4 CANNOT BE FACTORED OUT OF THIS TRINOMIAL. SO IF THIS DOES FACTOR, IT'S GOING TO FACTOR INTO TWO BINOMIAL FACTORS. AND FOR THE TRIAL AND ERROR METHOD, WE'LL CONSTRUCT OUR FACTORS BASED UPON THE FACT THAT THE FIRST TERMS OF THE BINOMIAL FACTORS MUST COME FROM THE FACTORS OF 4X SQUARED. AND THE SECOND TERMS OF OUR BINOMIAL FACTORS MUST COME FROM THE FACTORS OF - 15. SO WHAT WE'RE GOING TO DO IS USE A TRIAL AND ERROR PROCESS TO CONSTRUCT BINOMIAL FACTORS. AND NOW WE'LL CHECK TO SEE WHICH BINOMIAL FACTORS HAVE THE SUM OF THE INNER PRODUCT AND OUTER PRODUCT THAT'S EQUAL TO THE MIDDLE TERM OF -4X. SO FOR THE FACTORS OF 4X SQUARED WE COULD USE 2X AND 2X, BUT WE COULD ALSO USE 4X AND X. LET'S GO AHEAD AND SET THAT OPTION UP HERE BELOW. AGAIN, BECAUSE WE'RE USING THE TRIAL AND ERROR PROCESS, WE'RE NOT SURE WHICH ONE IS GOING TO WORK. AND NOW WE'RE GOING TO PLACE THE FACTORS OF -15 INTO THE SECOND POSITIONS, AND CHECK THE SUM OF THE INNER AND OUTER PRODUCT TO SEE WHICH WILL GIVE US A SUM OF -4X. WELL, THE FACTORS OF -15, WE COULD USE -3 x 5, -5 x 3, OR EVEN -1 x 15, OR -15 x 1. AGAIN, WE'RE NOT SURE WHICH OF THESE FACTORS ARE GOING TO WORK UNTIL WE TRY IT. SO FOR EXAMPLE, IF WE USE THE FACTORS -3 AND +5 WE COULD PUT A -3 HERE AND A +5 HERE. LET'S GO AHEAD AND DO THE SAME BELOW. AND NOW WE'LL CHECK THE SUM OF THE INNER AND OUTER PRODUCTS. SO HERE WE HAVE -3 x 2X THAT'S -6X + WE WOULD HAVE 2X x 5 THAT'S 10X. WELL, THIS GIVES US A SUM 4X AND WE WANT A SUM OF -4X. SO WE HAVE THE RIGHT NUMBER, WRONG SIGN. SO IT'S ACTUALLY GOING TO BE AN EASY FIX. WE JUST HAVE TO SWITCH THE SIGNS OF OUR FACTORS. SO WE'LL CHANGE THIS TO +3 AND WE'LL CHANGE THIS TO -5. AND NOW IF WE CHECK THIS, WE'LL HAVE +6X + -10X, WHICH DOES GIVE US -4X, OUR MIDDLE TERM. SO THIS IS HOW YOU GO ABOUT USING THE TRIAL AND ERROR METHOD TO FACTOR TRINOMIALS WHEN A ISN'T EQUAL TO 1. NOTICE HOW WE DIDN'T HAVE TO GO DOWN TO OUR SECOND OPTION OF USING 4X AND X. THESE ARE THE TWO BINOMIAL FACTORS OF THE GIVEN TRINOMIAL. NOW LETS TAKE A LOOK AT A SECOND EXAMPLE. NOTICE HOW THERE ARE NO COMMON FACTORS AMONG THESE THREE TERMS. SO THE FIRST TERMS OF THE BINOMIAL FACTORS MUST COME FROM THE FACTORS OF 20X SQUARED, AND THE SECOND TERMS OF THE BINOMIAL FACTORS MUST COME FROM THE FACTORS OF + 3. SO THE GOOD NEW IS THERE'S NOT VERY MANY FACTORS OF +3, BUT THERE ARE QUITE A FEW FACTORS OF 20X SQUARED. WE COULD USE 5X AND 4X, BUT WE COULD ALSO USE 10X AND 2X, AND WE COULD ALSO USE 20X AND X. AND THEN FOR THE FACTORS OF +3, SINCE THE SUM OF THE INNER AND OUTER PRODUCT HAS TO BE POSITIVE, WE'LL ONLY USE THE POSITIVE FACTORS OF 3. SO LETS TRY USING +3 AND +1. NOTICE HOW THE INNER PRODUCT IS 12X, AND THE OUTER PRODUCT IS 5X. THIS GIVES US THE SUM OF 17X AND WE WANT +19X, SO THAT'S NOT GOING TO WORK. LETS GO AHEAD AND TRY IT HERE, +3, +1. WELL, HERE WE HAVE 6X + THE OUTER PRODUCT IS 10X, THAT'S 16X, STILL NOT 19X. AND THIS ISN'T GOING TO WORK EITHER IF I PUT +3 HERE AND +1. NOTICE HOW WE HAVE 3X AND 20X THAT'S 23X. SO LET'S GO AHEAD AND SWITCH THE PLACEMENT OF OUR TWO FACTORS. LETS TRY +1 AND +3. BY THE WAY, WE KNOW IF WE'RE LOOKING FOR A SUM OF 19X, THERE'S NO WAY WE'RE GOING TO HAVE A FACTOR OF 20X SINCE ALL THE FACTORS ARE POSITIVE. SO WE WANT EVEN CHECK THIS ONE AGAIN. LETS CHECK THE FIRST ONE. THE INNER PRODUCT IS 4X, THE OUTER PRODUCT IS 15X THAT'S GOOD NEWS. THE SUM IS 19X, WHICH IS EQUAL TO OUR MIDDLE TERM. SO WE FOUND THE CORRECT FACTORS OF OUR TRINOMIAL USING TRIAL AND ERROR. WE HAVE THE QUANTITY OF 5X + 1 x THE QUANTITY 4X + 3. NOW, WE'RE GOING TO TAKE A LOOK AT THESE SAME TWO PROBLEMS USING THE GROUPING TECHNIQUE AS WELL AS USING A METHOD CALLED "BOTTOMS UP."