Answer:
The third degree polynomial function = x³ + 27x² + 200x + 300
Step-by-step explanation:
The third-degree polynomial function has zeros −2 and −15.
From the above, we have been given two factors of the polynomial function. Let's derive the factors from the two zeros of the polynomial given.
The two given zeros of the polynomial can be written as:
x= -2
x+2 = 0
(x+2) is a factor of the polynomial
x= -15
x+15 = 0
(x+15) is a factor of the polynomial
So we have two factors of the polynomial (x+2) and (x+15). But since it is a third degree polynomial, we have to find the third factor.
Let (x-b) be the third factor and f(x) represent the third degree polynomial
f(x) = (x-b) (x+2) (x+15)
Expanding (x+2) (x+15) = x² + 2x + 15x + 30
(x+2) (x+15) = x² + 17x + 30
f(x) = (x-b) (x² + 17x + 30)
From the given information, a value of 1,170 is obtained when x=3
f(3) = 1170
Insert 3 for x in f(x)
f(3) = (3-b) (3² + 17(3) + 30)
1170 = (3-b) (9 + 51 + 30)
1170 = (3-b) (90)
1170/90 = 3-b
3-b = 13
b = 3-13 = -10
Insert value of b in f(x)
f(x) = [x-(-10)] (x² + 17x + 30)
f(x) = (x+10) (x² + 17x + 30)
f(x) = x³ + 17x² + 30x + 10x² + 170x + 200x + 300
f(x) = x³ + 27x² + 200x + 300
The third degree polynomial function = x³ + 27x² + 200x + 300
