The following statement are equivalent r is a real root of a

The following statement are equivalent r is a real root of a poly\'l f r is an x-intercept of the graph of f x - r is a factor of f r is a solution to the equation f(x) = 0 Long distance of f(x) / x - r gives a reminder of 0

Solution

if r is real root of polynomial, it satisfies the equation so that f(r)=0

If we plot f(x), then x intercepts are the points where f(x)=0, f(r)=0

When r is a root of f(x), then f-r is one of the factor

Which is equivalent to f(r)=0, since it is a root of f(x), and it satisfies the f(r)=0

since x-r is one of the factor, f(x)/(x-r) should divide compeletely with reminder 0