- WE WANT TO FIND AN EQUATION FOR THE GRAPH
OF THE DEGREE 6 POLYNOMIAL FUNCTION
AND ALSO LEAVE THE FUNCTION IN FACTORED FORM.
BECAUSE THE POLYNOMIAL IS DEGREE 6
WE KNOW THERE IS AT MOST 6 REAL RATIONAL ZEROS OR ROOTS.
AND GRAPHICALLY, WE CAN FIND THESE
BY ANALYZING THE X INTERCEPTS OF OUR GRAPH.
SO LOOKING AT THE GRAPH
NOTICE HOW WE HAVE 1, 2, 3, 4 X INTERCEPTS,
THEREFORE THERE ARE FOUR REAL RATIONAL ZEROS OR ROOTS
BUT BECAUSE HOW THE FUNCTION BEHAVES
WE CAN SEE THAT SOME OF THE ROOTS
HAVE A MULTIPLICITY THAT'S GREATER THAN 1.
WHAT I MEAN BY THAT IS IF A FUNCTION CROSSES THE X AXIS
THEN THE ROOT OR ZERO HAS AN ODD MULTIPLICITY.
IF IT TOUCHES THE X AXIS
BUT DOESN'T CROSS THE X AXIS THE MULTIPLICITY IS EVEN.
SO NOTICE THE ZEROS AT (-2, +5) DO JUST TOUCH THE X AXIS
BUT DON'T CROSS THE X AXIS
WHICH MEANS THE ZEROS OF (-2, +5) HAVE AN EVEN MULTIPLICITY
AND BECAUSE WE HAVE FOUR X INTERCEPTS
WE COULD ASSUME BOTH HAVE A MULTIPLICITY OF 2.
SO IF THESE TWO HAVE MULTIPLICITY OF 2, 2, 4, 5, 6
WE HAVE ALL OF THE RATIONAL ZEROS
TO FIND THE EQUATION OF OUR FUNCTION.
TO DO THIS WE USE THE FACTORED FORM OF A POLYNOMIAL
GIVEN HERE BELOW WHERE "A" IS A CONSTANT
AND R SUB 1, R SUB 2, R SUB 3 AND SO ON
ARE THE ROOTS OR ZEROS OF THE POLYNOMIAL FUNCTION.
BUT TO DETERMINE THE VALUE OF "A"
WE'LL ALSO HAVE TO IDENTIFY ONE MORE POINT
ON THE FUNCTION.
LET'S GO AHEAD AND USE THE Y INTERCEPT OR THIS POINT HERE
WHERE THE COORDINATES ARE (0, -30).
SO WE'LL FIRST FIND THE FACTORS
BASED UPON OUR ROOTS OR ZEROS
AND THEN WE'LL FIND THE VALUE OF "A" USING THE POINT HERE.
SO WE'LL HAVE F OF X = "A" x--THE FIRST ZERO IS X = -3,
SO THE FACTOR MUST BE X - -3 OR X + 3.
THE NEXT ZERO IS -2,
SO WE'D HAVE A FACTOR OF X - - 2 OR X + 2
BUT BECAUSE THE MULTIPLICITY IS 2
WE NEED TWO FACTORS OF X + 2 OR THE QUANTITY X + 2 SQUARED.
NEXT WE HAVE A ZERO OF 1,
SO WE HAVE A FACTOR OF X - 1 AND OUR LAST ZERO IS X = 5,
SO WE HAVE A FACTOR OF X - 5
BUT AGAIN THIS HAS MULTIPLICITY 2,
SO WE HAVE TWO FACTORS OF X - 5
OR THE QUANTITY X - 5 SQUARED.
SO BASED UPON HOW OUR GRAPH BEHAVES THESE MUST BE FACTORS
OF OUR POLYNOMIAL FUNCTION.
AND NOW BECAUSE THE GRAPH CONTAINS THE POINT (0, -30)
THAT MEANS F OF 0 MUST EQUAL -30.
SO NOW WE'LL SUBSTITUTE 0 FOR X
AND SET THE FUNCTION VALUE = - 30
WHICH WILL ALLOW US TO SOLVE FOR "A".
SO F OF 0 WOULD BE EQUAL TO "A" x--
IF X IS 0 WE'D HAVE 3 HERE,
IF X IS 0 HERE WE'D HAVE 2 SQUARED.
IF X WAS 0 HERE WE'D HAVE -1 AND HERE WE'D HAVE -5X SQUARED
AND THIS MUST = -30.
SO NOW WE'LL GO AHEAD AND SIMPLIFY THIS
AND SOLVE FOR "A".
WE'D HAVE 3 x 4 THAT'S 12 x -1 WOULD BE -12 x +25
THAT'S GOING TO BE -300.
SO WE HAVE -300A = -30.
DIVIDE BOTH SIDES BY -300.
THIS SIMPLIFIES TO "A."
OVER HERE WE HAVE +1/10.
NOW THAT WE KNOW THE VALUE OF "A"
WE HAVE ALL THE INFORMATION WE NEED
TO FIND A FUNCTION TO REPRESENT OUR GRAPH.
WE'LL USE THIS FORM HERE AND SUBSTITUTE 1/10 FOR "A".
I THINK WE'LL ALSO CHANGE THE ORDER HERE
TO PUT THE SQUARED FACTORS TOWARD THE FRONT.
SO WE'LL HAVE F OF X = 1/10 x--
LET'S PUT THE QUANTITY X + 2 SQUARED FIRST.
AND THEN WE'LL PUT THE QUANTITY X - 5 SQUARED SECOND
AND THEN WE STILL HAVE A FACTOR OF X + 3
AND A FACTOR OF X - 1.
I HOPE YOU FOUND THIS HELPFUL.