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- WE WANT TO FIND ALL THE ZEROS OF THE FUNCTION
GIVEN ONE OF THE ZEROS IS +3I.
SO FIRST THING WE SHOULD RECOGNIZE HERE
IS THAT WE HAVE A DEGREE FIVE POLYNOMIAL,
SO WE SHOULD HAVE FIVE COMPLEX ZEROS.
NEXT, WE'RE GIVEN ONE ZERO AS 3I,
WHICH WE CAN THINK OF AS 0 + 3I,
AND SINCE COMPLEX ZEROS COME IN CONJUGATE PAIRS,
0 - 3I IS ALSO A ZERO OF THE FUNCTION.
SO ALREADY WE HAVE TWO OF THE FIVE ZEROS.
X = 3I AND X = -3I.
NEXT WE WANT TO FIND ALL OF THE REAL RATIONAL ZEROS
OF THIS FUNCTION.
TO DO THIS, WE COULD LIST ALL THE POSSIBLE RATIONAL ZEROS
BY FINDING ALL OF THE FACTORS OF 216
THEN DIVIDE THEM BY ALL OF THE FACTORS OF +2,
BUT THAT WOULD BE A VERY LONG LIST.
SO WHAT WE'RE GOING TO DO
IS ACTUALLY DETERMINE THE RATIONAL ZEROS
OF THIS FUNCTION BY ANALYZING THE GRAPH.
REMEMBER, THE ZEROS OF THIS FUNCTION ARE THE X VALUES
THAT MAKE THE FUNCTION EQUAL TO ZERO.
AND FUNCTION VALUES ARE OUR Y VALUES,
SO WE'RE LOOKING FOR THE VALUES OF X
WHERE Y IS EQUAL TO ZERO
WHICH MEANS THE X INTERCEPTS OF THIS FUNCTION
REPRESENT THE REAL ZEROS OF THE FUNCTION.
SO WE'RE GOING TO GRAPH THIS, LOOK AT THE X INTERCEPTS,
AND THEN FROM THE X INTERCEPTS
DETERMINE IF WE HAVE ANY REAL AND RATIONAL ZEROS.
SO LET'S TAKE A LOOK AT THE GRAPH,
HERE IS THE GRAPH OF OUR FUNCTION.
NOTICE HOW WE HAVE 1, 2, 3 X INTERCEPTS
WHICH MEANS WE HAVE THREE REAL ZEROS,
AND OF THESE THREE, WE WANT TO DETERMINE
IF ANY OF THEM ARE RATIONAL.
WELL, THIS FIRST ZERO AND THIS THIRD ZERO
DON'T LOOK LIKE THEY'RE GOING TO BE RATIONAL VALUES,
BUT THIS X INTERCEPT IN THE MIDDLE
DOES LOOK LIKE ITS X = 1.5 OR X = 3/2.
SO LET'S GO BACK TO THE PREVIOUS SCREEN,
AND WE'LL INCLUDE X = 3/2 AS ONE OF OUR ZEROS.
SO WE STILL NEED TO FIND TWO MORE ZEROS
WHICH WE KNOW ARE REAL.
NOW, FOR THE NEXT STEP, HAVING ZEROS OF THE FUNCTION
GIVE US INFORMATION ABOUT THE FACTORS OF THE FUNCTION
WHICH MEANS OUR FUNCTION, F OF X,
MUST HAVE A FACTOR OF X - 3I IF X = +3I IS A ZERO.
IF X = -3I IS A ZERO, THEN X + 3I MUST BE A FACTOR.
AND IF X = +3/2 IS A ZERO,
THEN X - 3/2 WOULD HAVE TO BE A FACTOR.
NOTICE HOW WE HAVE THREE DEGREE ONE FACTORS
AND OUR POLYNOMIAL IS A DEGREE FIVE,
WHICH MEANS WE'D BE LEFT WITH A DEGREE TWO FACTOR
WHICH WE CAN FIND BY DIVIDING THE FUNCTION
BY THESE LINEAR FACTORS.
SO NOW WE'RE GOING TO PERFORM DIVISION.
WE CAN USE EITHER LONG DIVISION
OR SYNTHETIC DIVISION.
LET'S GO AHEAD AND USE SYNTHETIC DIVISION.
WE'RE FIRST GOING TO DIVIDE BY THE FACTOR OF X - 3/2
BECAUSE IT'S EASIER TO PERFORM SYNTHETIC DIVISION
WITH OUR REAL RATIONAL ZEROS
EVEN THOUGH WE CAN STILL PERFORM SYNTHETIC DIVISION
WITH FACTORS CONTAINING COMPLEX OR IMAGINARY NUMBERS.
SO TO PERFORM SYNTHETIC DIVISION,
WE'RE GOING TO USE 3/2,
AND THEN WE'RE GOING TO LIST THE COEFFICIENTS
OF THE ORIGINAL POLYNOMIAL FUNCTION,
SO WE HAVE 2, -3, 2, -3, -144 AND +216.
WHEN WE DO THIS, REMEMBER, OUR REMAINDER HERE SHOULD BE ZERO.
SO WE BRING DOWN THE FIRST NUMBER WHICH IS 2.
2 x 3/2 IS 6/2 OR 3.
SO 0, 0 x 3/2 IS 0, BRING DOWN THE 2, 2 x 3/2 IS 3.
WE HAVE ANOTHER ZERO.
0 x 3/2 IS 0.
BRING DOWN THE -144.
NOW HERE, -144 x 3/2 IS EQUAL TO -216 WHICH IS GOOD NEWS.
WE HAVE A REMAINDER OF ZERO,
WHICH VERIFIES THIS IS A FACTOR OF OUR FUNCTION.
AND NOW THESE WOULD BE THE COEFFICIENTS
OF OUR DEGREE 4 FACTOR
WHICH WE'LL NOW DIVIDE BY THE FACTOR OF X - 3I.
SO NOW WE'LL USE 3I
AND PERFORM SYNTHETIC DIVISION AGAIN
USING THESE COEFFICIENTS.
SO BRING DOWN THE 2.
2 x 3I IS 6I. ADD 6I.
NOW 6I x 3I WOULD BE 18I SQUARED.
REMEMBER I SQUARED IS EQUAL TO -1,
SO THIS IS -18, ADD -16 x 3I IS -48I, ADD.
3I x -48I IS EQUAL TO -144I SQUARED,
BUT I SQUARED IS -1, SO THIS BECOMES +144,
WHICH AGAIN IS GOOD NEWS BECAUSE OUR REMAINDER IS ZERO.
NOW TO DIVIDE BY THE FACTOR OF X + 3I,
WE'LL USE -3I
AND PERFORM SYNTHETIC DIVISION AGAIN USING THESE COEFFICIENTS
WHICH WOULD BE THE COEFFICIENTS
OF THE DEGREE THREE FACTOR OF OUR FUNCTION.
SO WE BRING DOWN THE 2.
2 x -3I IS -6I. THIS IS ZERO.
0 x -3I IS 0, THAT GIVES US -16.
-3I x -16 IS +48I.
WE ADD, AND THIS WOULD BE ZERO.
SO BECAUSE WE DIVIDED BY A LINEAR FACTOR THREE TIMES,
THESE ARE THE COEFFICIENTS OF THE DEGREE TWO FACTOR,
WHICH MEANS THIS FACTOR HERE WOULD BE 2X SQUARED + 0X - 16.
NOW TO FIND THE TWO REMAINING ZEROS,
WE'RE GOING THE FIND THE ZEROS OF THIS DEGREE TWO FACTOR.
LET'S GO AHEAD AND DO THAT ON THE NEXT SLIDE.
NOW, BEFORE WE DO THIS,
NOTICE HOW THIS FACTOR HAS A COMMON FACTOR OF 2,
SO I'M GOING TO GO AHEAD AND WRITE THIS
AND I'LL FACTOR OUT THE 2.
BUT I'M GOING TO GO AHEAD AND PUT THE FACTOR OF 2
BEFORE THE FACTOR OF X - 3/2,
AND THEN TO CLEAR THIS FRACTION,
I CAN DISTRIBUTE THE 2.
NOTICE IF WE DISTRIBUTE THIS 2,
WE WOULD ELIMINATE THIS FRACTION HERE.
WE WOULD JUST HAVE A FACTOR OF 2X - 3.
AND AGAIN, OUR GOAL IS TO FIND THE X VALUES
WHERE THIS FUNCTION IS EQUAL TO ZERO.
WE FOUND THE ZEROS FROM THESE THREE FACTORS,
SO TO FIND THE REMAINING TWO,
WE'RE GOING TO SET THIS FACTOR HERE EQUAL TO ZERO.
SO WE NEED TO SET UP
AND SOLVE THE EQUATION X SQUARED - 8 = 0.
ADD EIGHT TO BOTH SIDES, SQUARE ROOT BOTH SIDES.
THIS WOULD BE PLUS OR MINUS,
SO WE HAVE X = +/- THE SQUARE ROOT OF 8.
WE CAN REWRITE THOUGH AS THE SQUARE ROOT OF 2 x 2 x 2.
IT HAS A PERFECT SQUARE FACTOR OF 4, SO IT SIMPLIFIES.
OUR TWO REMAINING ZEROS ARE X = +/- 2 SQUARE ROOT 2.
SO JUST TO SHOW THIS POLYNOMIAL
WITH FIVE LINEAR FACTORS,
WE CAN WRITE THIS AS F OF X = X - 3I, X + 3I, 2X - 3.
IF ONE OF THE ZEROS IS +2 SQUARE ROOT 2,
WE'D HAVE A FACTOR OF X - 2 SQUARE ROOT 2,
AND IF ONE OF THE ZEROS IS -2 SQUARE ROOT 2,
WE'D HAVE A FACTOR OF X + 2 SQUARE ROOT 2.
IF WE MULTIPLIED ALL OF THIS OUT,
WE'D END UP WITH THE ORIGINAL POLYNOMIAL FUNCTION.
BUT TO FINISH, AGAIN, OUR TWO REMAINING ZEROS
ARE X = 2 SQUARE ROOT 2, AND X = -2 SQUARE ROOT 2.
WHILE WE DID SHOW A LITTLE EXTRA WORK
TO SHOW THE POLYNOMIAL FUNCTION IN FACTORED FORM,
I THINK IT IS IMPORTANT TO MAKE THE CONNECTION
BETWEEN HAVING THE ZEROS OF A POLYNOMIAL FUNCTION
AND THE FACTORS OF A POLYNOMIAL FUNCTION.
I THINK IT'S ALSO IMPORTANT TO RECOGNIZE
WHAT THE GRAPH OF A POLYNOMIAL FUNCTION TELLS US
ABOUT THE ZEROS OF THE FUNCTION.
OKAY. I HOPE YOU FOUND THIS HELPFUL.