Question

I do find a polynomial such as `P(-3) = 0` `P(4) = 0` and `P'(-sqrt(3)) = 0` `P'(e) = 0` ?

Answer 1

`P(x) = x^4-1.89528x^3-8.55794x^2+8.19652x-30.5614`

Explanation:

Given four conditions, we will choose a four coefficient's polynomial.

`P(x) = x^4+ax^3+bx^2+c x+d`

Now, applying the conditions, with `e` the natural logarithms basis,

#{ (P(-3) = 81-a 27+b 9-c3=0), (P(4)=256+a64+b16+c4+d=0), (P'(-sqrt(3))=-4(sqrt(3))^3+3a(sqrt(3))^2-2bsqrt(3)+c=0), (P'(e)=4e^3+3e^2a+2eb+c=0) :}#

Solving for `a,b,c,d` we got

`{a = -1.89528, b = -8.55794, c = 8.19652, d = -30.5614}`