I need help explaining this theorem and example provided below.
$f (x) = (x^2 + bx + c)(x − d)$
$= x^3 + (2u − 3d)x^2 + (u^2 − 4ud + 3d^2 − v^2 )x− d(u − d − v)(u − d + v)$. (2)
For brevity, let us call polynomials with integer (respectively, rational) coefficients and roots Z-polynomials (respectively, Q-polynomials). The formula (2) describes all monic (leading coefficient 1) cubic Z-polynomials $ f$ with rational roots of $f'$ and $f''$.
To get the complete description of all cubic Q-polynomials $h$ with rational roots of $h'$ and $h''$ we need the following result of algebraic folk-lore: for a polynomial $f$ and number $α$ $\neq$ 0, define a new polynomial $f_{α}$ by $f_{a}(x)$ = $f(αx)$, then $r$ is a root of $f$ , if and only if $\frac{r}{a}$ is a root of $f_{α}$. Then the following proposition is the key to a full description:
Proposition & Proof to the Theorem: enter image description here
(1) $f (x) = x^3 − ux^2 − v^2x + uv^2 = (x − u) · (x^2 − v^2)$
Theorem. Let h be a cubic polynomial with rational coefficients. Then h, h' and h'' have rational roots, if and only if $h = μf_{λ}$ where μ, λ are rational numbers and $f$ is a polynomial of the form (1) with u, v.
For the practical application of this result, start with a polynomial $f$ of form (1) with u, v being integers. Then choose any rational λ,μ =/= 0 and form $μf_{λ}$. This is the polynomial to be graphed.
Example: For $u = 1, v = 4$, formula (4) gives $f (x) = x^3 − x^2 − 16x + 16$ with roots 1, 4, −4. For λ = 2, form $f_{2}$ and get $h(x) = \frac{1}{4} f_{2} = 2x^3 − x^2 − 8x + 4$ with roots $\frac{1}{2} , 2, −2$.
What happens if u = 13, v = 3 for formula (1)?