- WE WANT TO FACTOR AND SOLVE THE GIVEN POLYNOMIAL EQUATION.
REMEMBER, THE FIRST STEP IN FACTORING
IS TO FACTOR OUT THE GREATEST COMMON FACTOR.
NOTICE THE THREE TERMS ON THE LEFT
SHARE A COMMON FACTOR OF Z.
SO FOR THE FIRST STEP,
WE'LL FACTOR OUT Z FROM THE LEFT SIDE OF THE EQUATION.
SO WE'D HAVE Z x THE QUANTITY 20Z SQUARED - 11Z - 3 = 0.
NOTICE HOW WE FACTORED OUT ONE FACTOR OF Z FROM EACH TERM,
AND IF WE DISTRIBUTE,
WE STILL HAVE 20 Z CUBED - 11Z SQUARED - 3Z.
NOW WE WANT TO SEE IF WE CAN FACTOR THE TRINOMIAL
INSIDE THE PARENTHESES,
AND THIS IS ONE OF THE MORE CHALLENGING TYPES
OF TRINOMIALS TO FACTOR
BECAUSE THE LEADING COEFFICIENT HERE IS 20,
AND THERE ARE NO COMMON FACTORS AMONG THE THREE TERMS.
SO WE NEED TO FACTOR THE INNER TRINOMIAL
BY USING EITHER TRIAL AND ERROR, GROUPING,
OR BOTTOMS UP.
LET'S USE TRIAL AND ERROR.
SO WE STILL HAVE THIS FACTOR OF Z HERE,
AND NOW WE'RE GOING TO FACTOR THE INNER TRINOMIAL
INTO TWO BINOMIAL FACTORS.
NOW, USING TRIAL AND ERROR,
WE KNOW THAT THE TERMS IN THE FIRST POSITIONS,
OR THIS TERM AND THIS TERM MUST COME FROM THE FACTORS
OF 20Z SQUARED,
AND THE TERMS IN THE SECOND POSITIONS
MUST COME FROM THE FACTORS OF -3.
AND THEN THE SUM OF THE INNER PRODUCT AND OUTER PRODUCT
MUST GIVE US -11Z.
SO TO BEGIN, LET'S CONSIDER THE FACTORS OF 20Z SQUARED.
WELL, WE COULD USE 4Z AND 5Z, 2Z AND 10Z, OR 20Z AND 1Z.
LET'S GO AHEAD AND USE THE TWO FACTORS
THAT ARE CLOSEST TOGETHER, MEANING 5Z AND 4Z.
SO WE'LL HAVE A FACTOR OF 5Z HERE, AND A FACTOR OF 4Z HERE.
NOTICE THIS PRODUCT WOULD GIVE US 20Z SQUARED.
AND THEN LUCKILY THERE AREN'T A LOT OF FACTORS OF -3.
WE COULD -1 AND +3 OR -3 AND +1.
SO USING TRIAL AND ERROR,
WE'RE JUST GOING TO TRY TWO OF THESE,
PLACE THEM IN THE BINOMIALS
AND SEE IF THE INNER PRODUCT AND OUTER PRODUCT
DOES GIVE US A SUM OF -11.
SO FOR EXAMPLE, LET'S TRY THE -3 HERE, AND THE +1 HERE.
WELL, NOTICE HOW THE INNER PRODUCT IS -12Z
AND THE OUTER PRODUCT IS 5Z.
WELL, THIS SUM IS -7Z WHICH IS NOT -11Z,
WHICH MEANS WE'LL TRY SOMETHING ELSE.
LET'S GO AHEAD AND CLEAR THIS.
LET'S TRY SWITCHING THE FACTORS
AND PUTTING THE -3 HERE AND THE +1 HERE.
NOTICE HOW NOW THE OUTER PRODUCT IS -15Z
AND THE INNER PRODUCT IS +4Z,
AND NOTICE HOW WHEN WE ADD THESE
WE DO GET THE MIDDLE TERM IF -11Z
WHICH MEANS THIS IS NOW FACTORED CORRECTLY.
AND NOW THAT WE HAVE THIS FACTORED, WE CAN SOLVE.
NOTICE IN THIS CASE WE ACTUALLY HAVE THREE FACTORS.
WE HAVE A FACTOR OF Z, WE HAVE A FACTOR OF 5Z + 1,
AND WE HAVE A FACTOR OF 4Z -3,
WHICH MEANS ANY OF THESE FACTORS CAN EQUAL ZERO
AND THIS PRODUCT WOULD BE ZERO.
SO WE'RE GOING TO HAVE THREE SOLUTIONS.
EITHER THE FIRST FACTOR Z = 0,
OR THE SECOND FACTOR OF 5Z + 1 = 0,
OR THE THIRD FACTOR OF 4Z - 3 = 0.
AND NOW WE'LL SOLVE EACH OF THESE FOR Z.
NOTICE HOW THIS ONE'S ALREADY SOLVED FOR Z
SO THAT'S ONE SOLUTION.
TO SOLVE 5Z + 1 = 0 WE WOULD SUBTRACT 1 ON BOTH SIDES
AND THEN DIVIDE BY 5.
WE DIVIDE BY 5, NOTICE HOW Z IS EQUAL TO -1/5
WHICH IS OUR SECOND SOLUTION.
AND THEN FINALLY FOR 4Z - 3 = 0,
WE'D START BY ADDING 3 TO BOTH SIDES.
SO WE HAVE 4Z EQUALS +3, DIVIDE BOTH SIDES BY 4,
OUR THIRD SOLUTION IS Z = 3/4.
I HOPE YOU FOUND THIS HELPFUL.