+6 The polynomial f(x) has degree 3. If f(-1)=15, f(0)=0, f(1)=-5, and f(2)=12, then what are the x-intercepts of the graph of f(x)?
+128408 We have the form
y = ax^3 + bx^2 + cx + d
Because f(0) = 0.....then d = 0
And we have this system
a(-1)^3 + b(-1)^2 + c(-1) = 15
a(1)^3 + b(1)^2 + c(1) = 5
a(2)^3 + b(2)^2 + c(2) = 12
-a + b - c =15 (1)
a + b + c = 5 (2)
8a + 4b + 2c = 12 (3)
Add the first two equations and we get that 2b = 20 ⇒ b = 10
Multiply (2) by -2 ⇒ -2a -2b -2c = -10
Add this to (3)
6a + 2b = 2
6a + 2(10) = 2
6a = -18
a = -3
And using (2)
-3 + 10 + c = 5
c = -2
So....the polynomial is
f(x) = -3x^3 + 10x^2 - 2x
This graph shows the x intercepts : https://www.desmos.com/calculator/0qrtw2nob8
