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

Answer 1

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).