- WE WANT TO FIND ALL OF THE ZEROS OR ROOTS
OF THE GIVEN POLYNOMIAL FUNCTION.
TO DO THIS, WE HAVE TWO OPTIONS.
WE CAN GRAPH THE FUNCTION
AND TRY TO FIND THE REAL RATIONAL ZEROS FROM THE GRAPH,
OR WE CAN USE THE RATIONAL ROOTS THEOREM
THAT TELLS US THAT THE RATIONAL ZEROS
WOULD COME FROM THE RATIO OF THE FACTORS
OF THE CONSTANT TERM,
IN THIS CASE 15 OVER THE FACTORS
OF THE LEADING COEFFICIENT
WHICH, IN THIS CASE, WOULD BE 6.
BUT INSTEAD OF MAKING A LONG LIST
OF POSSIBLE RATIONAL ZEROS,
WE'RE ACTUALLY GOING TO TAKE A LOOK AT THE GRAPH
OF THIS FUNCTION TO DETERMINE OUR RATIONAL ZEROS.
SO HERE'S THE GRAPH OF OUR FUNCTION.
SINCE WE DO HAVE A CUBIC FUNCTION,
WHICH MEANS IT'S DEGREE 3, AND WE HAVE 3 X INTERCEPTS,
THIS TELLS US THAT WE DO HAVE THREE REAL ZEROS,
BUT WE'RE GOING TO USE A GRAPH
TO FIND THE REAL RATIONAL ZEROS.
NOTICE HOW THESE FIRST TWO ZEROS ON THE LEFT ARE REAL,
BUT THEY'RE EITHER FRACTIONAL OR IRRATIONAL,
SO LET'S GO AHEAD AND TAKE A LOOK
AT THIS THIRD ZERO OF X = +3.
WE'RE GOING TO USE THIS ZERO
TO HELP US FACTOR THE GIVEN FUNCTION
AND THEN FIND THE REMAINING TO ZEROS.
SO AGAIN, FROM THE GRAPH WE KNEW
THAT ONE OF THE ZEROS WAS X = +3.
WELL, IF WE HAVE A ZERO OF +3,
WE KNOW ONE OF THE FACTORS OF THE POLYNOMIAL FUNCTION
MUST BE X - 3.
SO LET'S WRITE THIS AS F OF X = X - 3
TIMES A DEGREE 2 FACTOR OR A QUADRATIC FACTOR.
SO TO FIND THIS FACTOR HERE,
WE CAN PERFORM LONG DIVISION BY DIVIDING BY X - 3,
OR WE COULD PERFORM SYNTHETIC DIVISION
BY USING THE X VALUE OF +3.
LET'S GO AND USE SYNTHETIC DIVISION.
SO WE'LL USE +3
AND THEN WE'LL LIST THE COEFFICIENTS OF OUR FUNCTION,
SO WE HAVE 6, -11, -26, AND 15.
NOW, WE'LL DIVIDE.
WE'LL BRING THE 6 DOWN, MULTIPLY BY 3, 6 x 3 IS 18,
ADD, MULTIPLY, 3 x 7 IS 21, ADD, MULTIPLY, -15.
THIS SUM IS ZERO WHICH IS GOOD.
THAT MEANS THE REMAINDER IS ZERO,
VERIFYING ALGEBRAICALLY 3 IS A 0 OF THE FUNCTION.
BUT THIS ALSO TELLS US THAT THIS QUADRATIC FACTOR
IS REPRESENTED BY THESE COEFFICIENTS HERE.
AGAIN, WE KNOW IT HAS TO BE A DEGREE 2 FACTOR,
SO THE OTHER FACTOR WOULD BE 6X SQUARE + 7X - 5.
AND AGAIN, WE'RE TRYING TO FIND THE X VALUES
THAT MAKE THIS EQUAL TO ZERO.
WE ALREADY HAVE THE 3 FROM THIS FACTOR.
NOW WE WANT O FIND THE ZEROS OF THIS QUADRATIC FACTOR.
TO DO THIS, WE EITHER CAN USE THE QUADRATIC FORMULA,
OR TRY TO FACTOR THIS.
AND BECAUSE GRAPHICALLY,
WE SAW THAT THE ZEROS OF THIS WILL BE REAL,
LET'S SEE IF WE CAN FACTOR IT.
AGAIN, IF IT ISN'T FACTORABLE,
WE'LL HAVE TO USE THE QUADRATIC FORMULA.
THIS IS A LITTLE BIT MORE CHALLENGING TO FACTOR
BECAUSE OF THE 6X SQUARED HERE.
WE DON'T KNOW IF IT'S GOING TO BE 6X AND X OR 3X AND 2X.
LET'S TRY 3X AND 2X.
AND THEN WE'LL PLACE THE FACTORS OF -5
IN THE SECOND POSITIONS
SO THAT SOME OF THE INNER PRODUCT AND OUTER PRODUCT
IS EQUAL TO +7.
SO IF WE PUT +5 HERE, WE'D HAVE TO PUT -1 HERE.
NOTICE HOW WE HAVE +10 HERE AND A -3 HERE,
WHICH DOES GIVE US THE SUM OF +7X.
SO IT DOES FACTOR,
AND, THEREFORE, THE REMAINING TWO ZEROS MUST BE
WHEN 3X + 5 = 0,
SO WE'D SUBTRACT 5 DIVIDE BY 3.
WE HAVE X = -5/3.
AND ALSO, WHEN 2X - 1 = 0,
ADD 1 TO BOTH SIDES, DIVIDE BY TWO, WE HAVE X = 1/2.
SO AGAIN, OUR THREE ZEROS ARE =3, -5/3, AND +1/2.
AGAIN, ALL THESE ARE REAL AND RATIONAL,
SO NOTICE THEY ALL COULD HAVE BEEN FOUND
BY LOOKING AT THE FACTORS OF 15
DIVIDED BY THE FACTORS OF 6.
BUT THERE WOULD OBVIOUSLY BE A LOT OF FACTORS
TO TRY TO FIND THESE THREE,
BUT IF WE GO BACK AND TAKE A LOOK AT THE GRAPH,
WE CAN PRETTY EASILY VERY THIS IS X = -5/3 OR - 1 2/3.
HERE'S OUR ZERO OF +1/2, AND OF COURSE OUR 0 OF +3.
OKAY. THAT'S GOING TO DO IT FOR THIS EXAMPLE.
WE WILL BE LOOKING AT SEVERAL MORE EXAMPLES
IN THE NEXT FEW VIDEOS.