- 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.