determine the polynomial f(x) that possesses the following characteristic :
1) f(x) is a polynomial of degree4
2) (x-1) is a factor of f(x) and f ' (x)
3) f(0) =3 and f'(0)= -5
4) the ramiander when f(x) is divided by (x-2) is 13
Find the set of value of x such tht f'(x) >0. find all four roots of f(x) = 0
f(x) = ax^4 + bx^3 + cx^2 +dx +e
f(1) = a+B+C+D+E = 0
f '(X) = 4ax^3+3b^2 +2cx + d = 0
f " (1) = 4a+3b+2c+d= 0
f(0)= e =3
f'(0) =d =-5
then wht to do?
2a)given that f(x) =x^n-nx+n-1 for the integer n>1. by considering f(x) and f'(x), show that f(x)=(X-1)^2 X g (x) is true for all polynomial g(x) with interger coefficients.
hence , or otherwise
1) show that 3^2n - 8n -1 is divisiblke by 64 for all intergers n>1
2) show that the equation X^4 -4x+3 = 0
i am not sure whether my step is correct. kinda blur with the question
a)i substitue n= 2 into f(2) = X^2 -2x+1 is identity to (X-1)^2 g(x)
g(x) = 1 <--- interger coefficients ?
1)do not have any idea
2) not sure. by substituting the value of 1 into the f(1) = 1 -4+3 =0, then i differentiate 4x^3-4 = 0 then again substitue to show the value?