La formule ${\displaystyle \Delta _{P}=a^{2n-2}\prod _{1\leq i<j\leq n}(x_{i}-x_{j})^{2}}$ est invariante par translation du polynôme.

Notons ${\displaystyle P=a\prod _{i=1}^{n}(X-x_{i})}$ et ${\displaystyle Q=b\prod _{i=n+1}^{n+m}(X-x_{i})}$.

${\displaystyle \Delta _{PQ}=(ab)^{2(m+n)-2}\prod _{1\leq i<j\leq m+n}(x_{i}-x_{j})^{2}}$ est bien le produit de :

• ${\displaystyle \Delta _{P}=a^{2n-2}\prod _{1\leq i<j\leq n}(x_{i}-x_{j})^{2}}$ ;
• ${\displaystyle \Delta _{Q}=b^{2m-2}\prod _{n<i<j\leq m+n}(x_{i}-x_{j})^{2}}$ ;
• ${\displaystyle \left(\operatorname {Res} (P,Q)\right)^{2}=\left(a^{m}b^{n}\prod _{1\leq i\leq n<j\leq m+n}(x_{i}-x_{j})\right)^{2}}$.

${\displaystyle \Delta _{P}={\frac {1}{a}}\operatorname {Res} (P,P')}$. En remplaçant, dans l'expression trouvée pour ${\displaystyle \operatorname {Res} (P,pX^{3}+qX^{2}+rX+s)}$ dans l'exercice 1-1 (ou 3-1), ${\displaystyle (p,q,r,s)}$ par ${\displaystyle (4a,3b,2c,d)}$, on en déduit :

${\displaystyle {\begin{aligned}\Delta _{P}&=a^{2}d^{4}+a\left[-2bcd^{3}+cd^{2}(4c^{2}-6bd)-d^{2}(8c^{3}-18bcd+12ad^{2})+e(16c^{4}-48bc^{2}d+18b^{2}d^{2}+32acd^{2})\right]\\&\;+3b^{3}d^{3}+c^{2}d^{2}(9b^{2}-16ac)-d^{3}(-27b^{3}+72abc-48a^{2}d)+e^{2}(81b^{4}-288ab^{2}c+192a^{2}bd+128a^{2}c^{2})\\&\;+bcd^{2}(-6bc+12ad)-bd^{2}(-12bc^{2}+18b^{2}d+8acd)+be(-24bc^{3}+54b^{2}cd+16ac^{2}d-60abd^{2})-cd^{2}(18b^{2}c-32ac^{2}-12abd)\\&\;+ce(36b^{2}c^{2}-54b^{3}d+96abcd-48a^{2}d^{2}-64ac^{3})+de(-54b^{3}c+144abc^{2}+36ab^{2}d-160a^{2}cd)-4b^{3}d^{3}\\&\;+4b^{2}\left[2c^{2}d^{2}-d^{2}(4c^{2}-6bd)+e(8c^{3}-18bcd+12ad^{2})\right]\\&\;+4b\left[-3bc^{2}d^{2}-d^{3}(9b^{2}-16ac)+e^{2}(-27b^{3}+72abc-48a^{2}d)-cd^{2}(-6bc+12ad)+ce(-12bc^{2}+18b^{2}d+8acd)+de(18b^{2}c-32ac^{2}-12abd)\right]\\&\;+16ac\left[-c^{2}d^{2}+3bd^{3}+ce(4c^{2}-6bd)+e^{2}(9b^{2}-16ac)+de(-6bc+12ad)\right]\\&\;+64a^{2}d\left[-d^{3}+2cde-3be^{2}\right]+256a^{3}e^{3}\\&\\&=256a^{3}e^{3}+a^{2}\left[(192-4\times 48-64\times 3)bde^{2}+(128-16\times 16)c^{2}e^{2}+(32-48-160+64\times 2+16\times 12)cd^{2}e+(1-12+48-64)d^{4}\right]\\&\;+a\left[(-288+4\times 72+16\times 9)b^{2}ce^{2}+(18-60+36+4\times 12-4\times 12)b^{2}d^{2}e+(-48+16+96+144+4(8-32)-16\times 6-16\times 6)bc^{2}de\right]\\&\;+a\left[(-2-6+18-72+12-8+12+4\times 16-4\times 12+16\times 3)bcd^{3}+(4-8-16+32-16)c^{3}d^{2}+(16-64+16\times 4)c^{4}e\right]\\&\;+b^{4}e^{2}\left[81-4\times 27\right]+b^{3}\left[(54-54-54-4\times 18+4\times 18+4\times 18)cde+(3+27-18-4+4\times 6-4\times 9)d^{3}\right]\\&\;+b^{2}c^{2}\left[(9-6+12-18+4(2-4-3+6))d^{2}+(-24+36+4(8-12))ce\right]\\&\\&=256a^{3}e^{3}+a^{2}(-192bde^{2}-128c^{2}e^{2}+144cd^{2}e-27d^{4})\\&\;+a(144b^{2}ce^{2}-6b^{2}d^{2}e-80bc^{2}de+18bcd^{3}+16c^{4}e-4c^{3}d^{2})+b^{2}(-27b^{2}e^{2}+18bcde-4bd^{3}+c^{2}(d^{2}-4ce)).\end{aligned}}}$

${\displaystyle \Delta _{P}=a^{2n-2}\prod _{1\leq i<j\leq n}(x_{i}-x_{j})^{2}}$.

${\displaystyle \Delta _{Q}=a^{2n-2}\prod _{1\leq i<j\leq n}\left({\frac {x_{i}}{\lambda }}-{\frac {x_{j}}{\lambda }}\right)^{2}={\frac {\Delta _{P}}{\lambda ^{n(n-1)}}}}$.

${\displaystyle \Delta _{P}=\prod _{1\leq i<j\leq n}(x_{i}-x_{j})^{2}}$.

${\displaystyle \Delta _{Q}=\prod _{1\leq i<j\leq n}\left({\frac {1}{x_{i}}}-{\frac {1}{x_{j}}}\right)^{2}={\frac {\Delta _{P}}{\prod _{i\neq j}x_{i}x_{j}}}={\frac {\Delta _{P}}{((-1)^{n}\lambda )^{n-1}}}={\frac {\Delta _{P}}{\lambda ^{n-1}}}}$.