f(x)=ax^3+bx^2+cx+d and r=root of f(x)=0.
(a)show that r-h is a root of f(x+h)=0
(b)show that -r is a root of f(-x)=0
(c)show that kr is a root of f(x/k)=0

Question

f(x)=ax^3+bx^2+cx+d and r=root of f(x)=0. 
(a)show that r-h is a root of f(x+h)=0
(b)show that -r is a root of f(-x)=0
(c)show that kr is a root of f(x/k)=0

anonymous · Answer

Let r be a root of the polynomial. Since r is a root, we know that:$$f(r)=0.$$I'll show part a), the others are done similarly. I want to show that r-h is a root of f(x+h).  So what happens if I plug in x = r-h?  We get:$$f(x+h)=f((r-h)+h)=f(r)$$and since f(r) = 0, we see that:$$f(x+h)=0$$when x = r-h. So r-h is a root of f(x+h).