Question

$F\left(x\right)$ is a polynomial of degree $3$ with rational coeffiecient . If its graph touches the x-axis, then how many roots are rational? Why?

Answer 1

We have,

$F\left(x\right)$ is a polynomial of degree $3$ with rational coefficient. And if graph touches the x-axis then one real root is guaranteed.

As it is a cubic polynomial it must have a real root.

Now if a function touches the x-axis then that particular point must be a repeated root of the function. So now we have two real roots (but no distinct). Since the other root can not be imaginary (Imaginary roots always occur in pairs), the other root must also be real. So that given the graph.

Say for example this graph.

The equation to the curve is thus

$(x-0)(x-0)(x+3)$

$=x^2(x+3)$

$=x^3+3x^2$

The polynomial must be the form ${(x-a)}^2\times (x-b)=0$.

Expanding this we get,

$x^3-(b+2a)x^2+x(x^2+2ab)-a^{2}b=0.$

So, $b+2a$ must be rational.

Hence this is the answer.