- WE WANT TO FIND THE EQUATION
OF A DEGREE 4 POLYNOMIAL FUNCTION
WITH REAL COEFFICIENTS HAVING ZEROS OR ROOTS
X = 4 WITH THE MULTIPLICITY 2,
X = -3, AND X = 0 WITH MULTIPLICITY 1.
THE FUNCTION ALSO PASSES THROUGH THE POINT (-2, -36).
SO TO FIND OUR POLYNOMIAL FUNCTION,
WE'RE GOING TO FIND IT IN FACTORED FORM
OR THE FORM GIVEN HERE BELOW WHERE F OF X = A, SOME CONSTANT
x THE FACTOR OF X - R SUB 1,
x THE FACTOR OF X - R SUB 2 AND SO ON
WHERE R SUB 1, R SUB 2, R SUB 3 ARE THE ZEROS
OR ROOTS OF THE FUNCTION.
REMEMBER THE ZEROS OR ROOTS ARE THE VALUES OF X
THAT MAKE THE FUNCTION EQUAL TO ZERO.
SO WE'LL START BY FINDING THE FACTORS
OF OUR POLYNOMIAL FUNCTION
AND THEN WE'LL FIND THE VALUE OF "A" USING THE GIVEN POINT.
SO WE'LL START WITH F OF X = "A" x THE FACTOR OF--
WELL THE FIRST ZERO IS +4,
SO OUR FACTOR WOULD HAVE TO BE X - 4.
NOTICE THIS ZERO HAS MULTIPLICITY 2,
WHICH MEANS X = 4 IS A DOUBLE ZERO
SO WE'LL HAVE 2 FACTORS OF X - 4
OR WE CAN JUST SAY THE QUANTITY X - 4 SQUARED.
NOTICE IF WE SUBSTITUTE 4 FOR X THIS FACTOR WOULD BE ZERO
MAKING THE FUNCTION VALUE EQUAL TO ZERO.
THE NEXT ZERO IS X = -3
SO WE'D HAVE TO HAVE A FACTOR OF X - -3 WHICH IS X + 3.
THIS HAS MULTIPLICITY 1, SO WE HAVE ONE FACTOR.
NOTICE IF WE SUBSTITUTE -3 FOR X THIS FACTOR WOULD BE ZERO
AND OUR LAST ZERO IS X = 0 SO WE'D HAVE A FACTOR OF X - 0.
NOW LET'S GO AHEAD AND CLEAN THIS UP.
X - 0 IS JUST X
SO WE CAN WRITE THIS AS "AX" x THE QUANTITY X - 4 SQUARED
x THE QUANTITY X + 3.
NOW LET'S FIND THE VALUE OF "A".
WE CAN DO THIS USING THE GIVEN POINT HERE.
IF THE FUNCTION CONTAINS THE POINT (-2, -36)
THAT MEANS F OF -2 MUST EQUAL -36.
SO LET'S FIND F OF -2 BY SUBSTITUTING -2 FOR X
AND THEN WE'LL SET THIS EQUAL TO -36 TO FIND THE VALUE OF "A".
SO WE WOULD HAVE "A" x -2 OR -2A x--
THIS WOULD BE -2 - 4 THAT'S -6 SQUARED
AND HERE WE'D HAVE -2 + 3 THAT'S +1.
AND THIS MUST EQUAL -36.
SO WE HAVE -2 x "A" x +36 x 1.
THAT'S GOING TO BE -72A = -36 DIVIDE BOTH SIDES BY -72
AND WE HAVE "A" EQUALS +1/2.
SO NOW THAT WE HAVE EVERYTHING IN FACTORED FORM
OUR POLYNOMIAL FUNCTION WILL BE F OF X EQUALS--
LET'S GO AHEAD AND USE THIS FORM HERE.
SO WE'LL HAVE 1/2X x THE QUANTITY X - 4 SQUARED
x THE FACTOR OF X + 3.
THIS WOULD BE OUR DEGREE 4 POLYNOMIAL FUNCTION
THAT HAS THE GIVEN CHARACTERISTICS.
LET'S GO AHEAD AND VERIFY THIS GRAPHICALLY.
HERE'S OUR ZERO OR ROOT OF -3 AND 0,
NOTICE HOW THE MULTIPLICITY IS ODD OR EQUAL TO 1
SO THE FUNCTION CROSSES THE X AXIS
AND THEN OVER HERE AT +4
NOTICE HOW THE FUNCTION ONLY TOUCHES THE X AXIS,
IT DOESN'T INTERSECT THE X AXIS.
THIS IS HOW A FUNCTION BEHAVES WHEN THE MULTIPLICITY IS EVEN.
SO THE ZEROS OR ROOTS OF THIS FUNCTION LOOK GOOD
AND AT THE LAST VERIFICATION
LET'S MAKE SURE THIS FUNCTION CONTAINS THE POINT (-2, -36).
WELL HERE'S OUR POINT WHEN X = -2
AND THE Y COORDINATE OR FUNCTION VALUE DOES LOOK LIKE
IT'S EQUAL TO -36.
SO THIS GRAPH VERIFIES THAT OUR WORK IS CORRECT.
I HOPE YOU FOUND THIS HELPFUL.