- WE WANT TO FIND AN EQUATION FOR THE GRAPH
OF THE DEGREE 4 POLYNOMIAL FUNCTION
AND LEAVE THE FUNCTION IN FACTORED FORM.
BECAUSE WE HAVE A DEGREE 4 POLYNOMIAL FUNCTION,
WE SHOULD RECOGNIZE
THAT WE HAVE AT MOST FOUR REAL RATIONAL ZEROS OR ROOTS
OF THE FUNCTION.
BUT LOOKING THE GRAPH,
WE CAN SEE THAT ALL ROOTS OR ZEROS ARE RATIONAL
BECAUSE THIS GRAPH HAS FOUR X INTERCEPTS.
THE X INTERCEPTS ARE (-3, -1) (+2, +5)
AND THESE ARE ALSO THE ROOTS OR ZEROS
OF OUR POLYNOMIAL FUNCTION.
SO BECAUSE WE KNOW THE ROOTS OR ZEROS OF OUR FUNCTION
WE CAN WRITE THE POLYNOMIAL FUNCTION IN FACTORED FORM
USING THIS FORM HERE
WHERE "A" IS THE CONSTANT AND R SUB 1, R SUB 2, R SUB 3
AND SO ON ARE THE ROOTS OR ZEROS OF THE FUNCTION.
BUT WE'RE ALSO GOING TO HAVE TO FIND ONE MORE POINT
OF THE FUNCTION TO DETERMINE THE VALUE OF "A".
SO LET'S GO AHEAD AND USE THE Y INTERCEPT HERE.
THE Y INTERCEPT IS -15
SO THE COORDINATES OF THIS POINT WOULD BE (0, -15).
SO TO SET THIS UP,
WE'LL FIRST FIND THE FACTORS OF OUR POLYNOMIAL FUNCTION
FROM THE ZEROS OF THE FUNCTION
AND THEN WE'LL USE THE POINT HERE
TO DETERMINE THE VALUE OF "A".
SO WE'LL HAVE F OF X = "A" x-- IF ONE OF THE ZEROS IS X = -3
THEN THE FACTOR MUST BE X - -3 OR X + 3.
IF ONE OF THE FACTORS IS X = -1,
THEN A FACTOR MUST BE X - -1 OR X + 1.
NEXT, IF WE HAVE A ZERO OF +2
WE MUST HAVE A FACTOR OF X - 2.
AND IF WE HAVE A ZERO OF +5 WE MUST HAVE A FACTOR OF X - 5.
NOTICE HOW THE CONSTANTS IN OUR FACTORS
HAVE THE OPPOSITE SIGN OF THE ZEROS FROM OUR GRAPH.
IF WE SUB THESE VALUES INTO OUR FUNCTION,
ONE OF THESE FACTORS WOULD BE ZERO
THEREFORE THIS PRODUCT WOULD BE ZERO
AND THE FUNCTION WOULD EQUAL ZERO INDICATED BY THE GRAPH.
AND NOW TO FIND THE VALUE OF "A" WE'LL USE THIS POINT HERE.
IF THE FUNCTION CONTAINS THE POINT (0, -15)
THIS MEANS F OF 0 MUST EQUAL -15.
SO WE'LL SUBSTITUTE ZERO FOR X
AND SET THIS FUNCTION VALUE EQUAL TO -15.
SO IF X IS EQUAL TO ZERO
WE WOULD HAVE ("A" x 3) x (1 x -2) x -5
AND THIS PRODUCT MUST EQUAL -15.
SO NOW WE'LL GO AHEAD AND SOLVE FOR "A".
(3 x 1) x -2 THAT'S -6 x -5 IS +30.
SO WE HAVE 30A = -15, DIVIDE BOTH SIDES BY 30.
WE HAVE "A" EQUALS-- THIS SIMPLIFIES TO -1/2.
SO NOW WE HAVE ALL THE INFORMATION WE NEED
TO WRITE OUR POLYNOMIAL FUNCTION IN FACTORED FORM.
USING THIS FORM OF OUR FUNCTION
WE'LL SUBSTITUTE -1/2 FOR "A".
SO WE'D HAVE F OF X = -1/2 x THE QUANTITY X + 3
x THE QUANTITY X + 1 x THE QUANTITY X - 2
AND x THE QUANTITY X - 5.
NOTICE IF WE MULTIPLY THIS OUT
THE LEADING COEFFICIENT WOULD BE NEGATIVE
AND IT'S DEGREE 4 POLYNOMIAL FUNCTION
SO NOTICE AS WE MOVE TO THE LEFT AND RIGHT
THE FUNCTION IS GOING DOWN APPROACHING NEGATIVE INFINITY
IN BOTH DIRECTIONS
WHICH IS WHAT WE EXPECT WITH AN EVEN DEGREE
AND A NEGATIVE LEADING COEFFICIENT.
OKAY. I HOPE YOU FOUND THIS HELPFUL.
WE'LL TAKE A LOOK AT ANOTHER EXAMPLE IN THE NEXT VIDEO.