A polynomial of degree 3 that has three distinct x-intercepts and whose graph rises to the left and falls to the right

Question

write a polynomial with the given charactistics

Answer 1

Suppose the 3 distinct x-intercepts are r₁, r₂, and r₃.  Then the function may be expressed as

 

f(x) = a(x - r₁)(x - r₂)(x - r₃),

 

with a < 0.

 

The a < 0 condition makes it approach ∞ as x approaches -∞, and makes it approach -∞ as x approaches ∞ (i.e. rises to the left, falls to the right).  You can expand f(x) so it has the more familiar polynomial form of 

 

f(x) = ax³ + bx² + cx + d,

 

with a < 0.