- NOW WE WANT TO FACTOR TRINOMIALS
IN THE FORM OF AX SQUARED + BX + C WHEN A IS = TO 1,
MEANING THE FIRST TERM IS GOING TO BE X SQUARED.
SO LOOKING AT OUR FIRST EXAMPLE,
THE ONLY COMMON FACTOR BETWEEN THESE THREE TERMS IS 1.
SO IF THIS DOES FACTOR,
IT'S GOING TO FACTOR INTO TWO BINOMIALS.
AND THE FIRST TERMS OF EACH BINOMIAL
WILL COME FROM THE FACTORS OF X SQUARED.
AND THE SECOND TERMS IN THE BINOMIAL FACTORS
WILL COME FROM THE FACTORS OF +32.
BUT WE DON'T WANT JUST ANY FACTORS OF 32.
WE WANT THE FACTORS OF 32 THAT ADD TO THE MIDDLE COEFFICIENT,
IN THIS CASE +12.
SO WE NORMALLY SAY WE WANT THE FACTORS OF C THAT ADD TO B.
WELL, THERE'S ONLY TWO FACTORS OF X SQUARED,
THAT WOULD BE X AND X.
SO THE MAIN CHALLENGE FOR THIS PROBLEM
IS TO FIND THE FACTORS OF +32 THAT ADD TO +12.
LET'S LIST THE POSITIVE FACTORS OF 32.
WE COULD HAVE 1 x 32, 2 x 16, 4 x 8,
BUT, AGAIN, WE WANT THE FACTORS THAT ADD TO 12.
SO 4 AND 8 ARE THE WINNING FACTORS.
AND SINCE 4 AND 8 ARE BOTH POSITIVE,
WE'LL HAVE +4 HERE AND +8 HERE.
OF COURSE WE COULD PUT THE +8 HERE AND THE +4 HERE
AND THE FACTORS WOULD BE THE SAME,
JUST IN A DIFFERENT ORDER.
IF WE WANT TO VERIFY THIS, WE CAN MULTIPLY THIS OUT.
WE'D HAVE X x X THAT WOULD X SQUARED.
AND THEN WE'D HAVE X x 8 THAT'S +8X.
THEN WE HAVE 4 x X THAT'S +4X.
AND THEN WE HAVE 4 x 8 THAT'S +32.
SO WE DO HAVE X SQUARED + 12X + 32.
SO ALL THESE FACTORING PROBLEMS CAN BE VERIFIED
BY DETERMINING THE PRODUCT OF OUR FACTORS,
BUT THIS WOULD BE OUR TRINOMIAL IN FACTORED FORM.
LET'S TAKE A LOOK AT ANOTHER EXAMPLE.
THE FIRST THING I NOTICED HERE,
THE ONLY COMMON FACTOR AMONG THESE TERMS IS 1.
SO IF THIS DOES FACTOR, IT'LL FACTOR INTO TWO BINOMIALS.
THE FIRST TERMS WILL COME FROM THE FACTORS OF X SQUARED
THAT WOULD BE X AND X.
THE SECOND TERMS WILL COME FROM THE FACTORS OF -60,
BUT WE WANT THE FACTORS OF -60 THAT ADD TO +11.
LET'S LIST SOME OF THE FACTORS OF -60
AND SEE IF WE CAN FIND THE PAIR THAT HAVE A SUM OF 11.
WELL, WE'D HAVE -1 x 60, -2 x 30, -3 x 20,
-4 x 15, -5 x 12, -6 x 10,
AND LET'S SEE ANY OF THESE HAVE A SUM OF +11.
WELL, -4 + 15 WOULD HAVE A SUM OF +11,
SO THESE ARE THE WINNING FACTORS.
SINCE ONE FACTOR IS -4 WE'LL HAVE X - 4.
AND SINCE ONE FACTOR IS +15 WE'LL HAVE + 15.
HERE'S OUR TRINOMIAL IN FACTORED FORM.
IN OUR NEXT EXAMPLE,
AGAIN, THE ONLY COMMON FACTOR AMONG THESE THREE TERMS
WOULD BE +1.
SO IF THIS DOES FACTOR, IT'LL FACTOR INTO TWO BINOMIALS.
THE FACTORS OF X SQUARED, X x X.
AND NOW WE WANT THE FACTORS OF +20 THAT ADD TO -9.
SO THEY WANT THE FACTORS OF +20 THAT ADD TO -9.
NOTICE BOTH FACTORS WILL HAVE TO BE NEGATIVE.
SO WE HAVE -1 x -20, -2 x -10, -4 x -5,
AND -4 + -5 IS = TO -9,
OUR MIDDLE TERM, OUR MIDDLE COEFFICIENT.
SO WE'RE GOING TO HAVE X - 4 x X - 5.
SO HERE'S A TRINOMIAL IN FACTORED FORM.
IN THE LAST EXAMPLE WE HAVE X SQUARED - 5X - 24.
THE ONLY COMMON FACTOR AMONG THESE THREE TERMS IS 1.
SO, AGAIN, IF THIS DOES FACTOR,
IT'LL FACTOR INTO TWO BINOMIALS.
THE FACTORS OF X SQUARED WOULD BE X AND X.
AND NOW WE WANT THE FACTORS OF -24 THAT ADD TO -5.
LET'S TRY -1 x 24, -2 x 12, -3 x 8.
I THINK I'LL STOP HERE BECAUSE NOTICE THAT -3 + 8 WILL BE +5,
AND SINCE WE WANT A -5,
WE CAN JUST CHANGE THE SIGNS OF THESE FACTORS.
IF WE USE A -8 AND A +3, THESE TWO WOULD HAVE A SUM OF -5.
SO WE HAVE X - 8 x THE QUANTITY X + 3.
AND NOW IT'S FACTORED.
HOPE YOU FOUND THIS HELPFUL.