- WE WANT TO FIND ALL THE ZEROS OF THE POLYNOMIAL FUNCTION
GIVEN 4I AS A ZERO.
THEN WE WANT TO WRITE P OF X AS A PRODUCT OF LINEAR FACTORS.
SO FOR A QUICK REVIEW,
THE COMPLEX FACTORIZATION THEOREM STATES
THAT IF WE HAVE A DEGREE N POLYNOMIAL FUNCTION,
THEN THERE ARE EXACTLY N COMPLEX ZEROS
AS LONG AS WE COUNT MULTIPLICITY.
SO WE'RE GOING TO FIND ALL THE COMPLEX ZEROS
OF OUR GIVEN POLYNOMIAL FUNCTION,
AND THEN USING THOSE ZEROS,
WE CAN WRITE THE POLYNOMIAL FUNCTION AS OUR PRODUCT
OF LINEAR FACTORS AS WE SEE HERE,
WHERE THE R'S WOULD BE THE ZEROS,
AND THE EXPONENTS M WOULD BE THE MULTIPLICITY,
AND IN THIS FORM, A WOULD BE A LEADING COEFFICIENT.
SO GOING BACK TO OUR EXAMPLE,
THE FIRST THING WE SHOULD NOTICE IS THAT WE'RE GIVEN 4I AS A 0.
SO WE'RE KEEPING A LIST OF OUR ZEROS.
WE KNOW THAT X = 4I AS A ZERO,
AND COMPLEX ZEROS ALWAYS COME IN PAIRS.
IF 0 + 4I IS A COMPLEX 0, WHICH WE'RE GIVEN,
THEN THE CONJUGATE OF THIS,
0 - 4I WOULD ALSO BE A 0.
SO WE CAN ALSO MAKE A NOTE THAT X = -4I
WOULD BE A ZERO OF THE DEGREE 5 POLYNOMIAL FUNCTION.
SO NOW WE WANT TO FIND 3 MORE ZEROS
SO WE CAN WRITE THE 5 LINEAR FACTORS
THAT WOULD MAKE UP THIS DEGREE 5 POLYNOMIAL FUNCTION.
TO DO THIS WE COULD USE THE RATIONAL ROOTS THEOREM,
THAT MAKING A LIST OF POSSIBLE RATIONAL ZEROS
USING A RATIO OF THE FACTORS OF THE CONSTANT TERM 64
OVER THE FACTORS OF THE LEADING COEFFICIENT 3.
BUT INSTEAD OF DOING THIS,
WE'RE GOING TO GRAPH THE POLYNOMIAL FUNCTION
AND LOOK FOR THE REAL RATIONAL X INTERCEPTS,
WHICH WOULD ALSO BE THE ZEROS OF P OF X.
SO HERE'S THE GRAPH OF THE GIVEN FUNCTION.
NOTICE HOW THERE ARE 3X INTERCEPTS.
SO THE 3 REMAINING ZEROS ARE REAL.
NOTICE 2 OF THESE ARE PRETTY EASY TO FIND.
HERE'S ONE X INTERCEPT AT -1, 0,
AND HERE'S ANOTHER X INTERCEPT AT 2, 0.
NOTICE AT THESE X VALUES,
THE Y VALUE OR THE FUNCTION VALUE WOULD BE 0.
THE ONE WE'RE NOT SURE ABOUT IS THIS ONE HERE.
WE PROBABLY COULD USE OUR GRAPHING CALCULATOR
TO DETERMINE THAT VALUE,
BUT WE'RE GOING TO GO AHEAD AND USE THE KNOWN ZEROS,
WHICH WOULD GIVE US FACTORS OF THE POLYNOMIAL FUNCTION
AND THEN PERFORM DIVISION TO FIND THIS REMAINING 0
AND REMAINING FACTOR.
SO GOING BACK TO THE PREVIOUS SLIDE,
LET'S ALSO RECORD THAT X = -1
AND X = 2 WOULD BE A 0 OF THE FUNCTION,
AND WE'RE STILL LOOKING FOR ONE MORE.
BUT BECAUSE WE HAVE FOUR ZEROS,
WE DO HAVE FOUR LINEAR FACTORS THAT WE CAN USE
TO HELP US FIND THE 5th FACTOR AND THIS 5th 0.
WE KNOW NOW THAT P OF X WILL HAVE 5 LINEAR FACTORS.
SO STARTING WITH X = 2, IF X = 2 IS A 0, X - 2
MUST BE A FACTOR.
NOTICE HOW X = 2 WOULD MAKE THIS FACTOR = 0,
AND THEREFORE, THIS FUNCTION VALUE = 0.
IF X = -1 IS A 0, THEN X - -1 OR X + 1 WOULD BE A FACTOR.
IF X = -4I IS A 0, THEN X - -4I OR X + 4I WOULD BE A 0,
AND IF X = 4I IS A 0, X - 4I WOULD BE A FACTOR.
SO IF ALL OF THESE ARE FACTORS
OF THE GIVEN POLYNOMIAL FUNCTION,
THEN THEY WOULD DIVIDE EVENLY INTO THE POLYNOMIAL FUNCTION.
SO WE CAN PERFORM DIVISION 1, 2, 3, 4 TIMES,
AND THAT WOULD GIVE US THE REMAINING LINEAR FACTOR,
WHICH WILL HELP US FIND THIS FACTOR AND THE LAST 0.
SO LET'S BEGIN BY DIVIDING THIS DEGREE 5
POLYNOMIAL BY THIS FIRST LINEAR FACTOR,
X - 2.
SO IF WE WANT TO DIVIDE THIS BY X - 2,
LET'S GO AHEAD AND USE SYNTHETIC DIVISION.
WE'LL START WITH THE 0 OF 2,
AND THEN WE'LL LIST THE COEFFICIENTS,
WHICH WILL BE 3, -5, 44, -76, -64, AND 64.
GO AHEAD AND BRING DOWN THE 3
TO BEGIN AND START MULTIPLYING BY 2.
2 x 3 = 6, AND THEN WE ADD.
2 x 1 = 2, AND WE ADD.
2 x 46 WOULD BE 92, AND WE ADD.
2 x 16 = 32, AND WE ADD.
2 x -32 = -64, AND WE ADD.
NOTICE HOW THE REMAINDER IS ZERO,
AND THAT'S BECAUSE X - 2
IS A FACTOR OF THIS POLYNOMIAL FUNCTION.
AND NOW, WE'RE GOING TO GO AHEAD
AND DIVIDE BY THE NEXT FACTOR OF X + 1.
SO IF WE WANT TO DIVIDE BY X + 1,
WE'LL ACTUALLY USE -1 TO PERFORM SYNTHETIC DIVISION.
THEN WE'LL GO AHEAD AND USE THESE VALUES HERE,
IGNORING THE REMAINDER.
BRING DOWN THE 3 AND START MULTIPLYING BY -1 THIS TIME.
-1 x 3 = -3, AND WE ADD.
IT'S - 2. -1 x -2 = 2.
WE ADD.
THAT'S 48. -1 x 48 = -48, AND WE ADD. THAT'S -32.
-1 x -32 = 32.
WE ADD, WHICH IS 0, WHICH IS GOOD NEWS,
AND NOW WE HAVE TWO MORE REMAINING FACTORS
THAT WE KNOW OF, X + 4I AND X - 4I.
AND WE CAN STILL PERFORM SYNTHETIC DIVISION
USING AN IMAGINARY NUMBER OR A COMPLEX NUMBER.
SO LET'S GO AHEAD AND TRY IT.
NOW, WE'RE GOING TO DIVIDE BY THE QUANTITY X + 4I.
SO WE'LL USE -4I HERE.
IT'LL BE A LITTLE MESSY, BUT WE CAN STILL DO IT.
REMEMBER, THE REMAINDER HERE STILL SHOULD BE ZERO.
SO BRING DOWN THE 3.
MULTIPLY BY -4.
THAT'S -12I.
WHEN WE ADD THIS, NOTHING SIMPLIFIES.
SO WE HAVE -2 - 12I.
WE'RE GOING TO MULTIPLY THIS BY -4I.
LET'S MULTIPLY THE TWO I TERMS FIRST.
-4I x -12I WOULD BE 48I SQUARED OR -48,
AND THEN WE HAVE -4I x -2I.
THAT'S + 8I.
WHEN WE ADD HERE, THOSE ARE OPPOSITES.
SO WE'RE LEFT WITH 8I, AND WE MULTIPLY THIS BY -4I.
IT'S -32I SQUARED OR 32.
ADD. THAT'S ZERO, AND WE'RE DONE.
NOTICE HOW, AGAIN, THE REMAINDER IS 0,
AND NOW, FOR THE LAST STEP, WE'RE GOING TO DIVIDE BY X - 4I.
SO WE'LL USE 4I FOR OUR SYNTHETIC DIVISION.
BRING DOWN THE 3.
4I x 3 = 12I.
NOTICE HOW THE I TERMS ARE OPPOSITES, GIVES US -2,
AND 4I x -2 = -8I.
AGAIN, WE HAVE A REMAINDER OF 0.
SO THIS REPRESENTS OUR LAST LINEAR FACTOR OF 3X - 2.
SO THIS LINEAR FACTOR WOULD BE 3X - 2,
AND THE 0 OF 3X - 2 = 2/3.
ADD 2 TO BOTH SIDES AND DIVIDE BY 3.
SO THE LAST 0 THAT WE'RE LOOKING FOR UP HERE IS 2/3.
SO BY USING THE GIVEN COMPLEX 0 AND ITS CONJUGATE
AND ANALYZING THE GRAPH OF THE POLYNOMIAL FUNCTION,
WE WERE ABLE TO FIND MOST OF THE ZEROS,
AND THEN WE FOUND THE LAST 0 BY PERFORMING DIVISION
USING THE KNOWN FACTORS AND THE KNOWN ZEROS.
SO WE FOUND ALL 5 COMPLEX ZEROS
OF THE GIVEN POLYNOMIAL FUNCTION,
AND WE ALSO FOUND THE 5 LINEAR FACTORS
OF THE GIVEN POLYNOMIAL FUNCTION.
I HOPE YOU FOUND THIS HELPFUL.