✅ Após resolver os cálculos, concluímos que o polinômio procurado é:
[tex]\Large\displaystyle\text{$\begin{gathered}\boxed{\boxed{\:\:\:\tt P(m, n) = 12mn - 6m^{2} - 6n^{2} + 10m^{2}n^{2}\:\:\:}}\end{gathered}$}[/tex]
Sejam os polinômios:
[tex]\Large\begin{cases} P(m, n) = \:?\\Q(m, n) = 5m^{2} - 8mn + 9n^{2} - 6m^{2}n^{2}\\S(m, n) = 4mn - m^{2} + 3n^{2} + 4m^{2}n^{2}\end{cases}[/tex]
Então, temos:
[tex]\Large\displaystyle\text{$\begin{gathered} \bf(I)\end{gathered}$}[/tex] [tex]\Large\displaystyle\text{$\begin{gathered} S(m, n) = P(m, n) + Q(m, n)\end{gathered}$}[/tex]
Isolando "P(m, n)" no primeiro membro da equação "I", temos:
[tex]\large\displaystyle\text{$\begin{gathered} P(m, n) = S(m,n) - Q(m,n)\end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered} = (4mn - m^{2}+ 3n^{2} + 4m^{2}n^{2}) - (5m^{2} - 8mn + 9n^{2} - 6m^{2}n^{2})\end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered} = 4mn - m^{2} + 3n^{2} + 4m^{2}n^{2} - 5m^{2} + 8mn - 9n^{2} + 6m^{2}n^{2}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = 12mn - 6m^{2} - 6n^{2} + 10m^{2}n^{2}\end{gathered}$}[/tex]
Portanto, o polinômio é:
[tex]\Large\displaystyle\text{$\begin{gathered} P(m, n) = 12mn - 6m^{2} - 6n^{2} + 10m^{2}n^{2}\end{gathered}$}[/tex]
[tex]\LARGE\displaystyle\text{$\begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered}$}[/tex]
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✅ Após resolver os cálculos, concluímos que o polinômio procurado é:
[tex]\Large\displaystyle\text{$\begin{gathered}\boxed{\boxed{\:\:\:\tt P(m, n) = 12mn - 6m^{2} - 6n^{2} + 10m^{2}n^{2}\:\:\:}}\end{gathered}$}[/tex]
Sejam os polinômios:
[tex]\Large\begin{cases} P(m, n) = \:?\\Q(m, n) = 5m^{2} - 8mn + 9n^{2} - 6m^{2}n^{2}\\S(m, n) = 4mn - m^{2} + 3n^{2} + 4m^{2}n^{2}\end{cases}[/tex]
Então, temos:
[tex]\Large\displaystyle\text{$\begin{gathered} \bf(I)\end{gathered}$}[/tex] [tex]\Large\displaystyle\text{$\begin{gathered} S(m, n) = P(m, n) + Q(m, n)\end{gathered}$}[/tex]
Isolando "P(m, n)" no primeiro membro da equação "I", temos:
[tex]\large\displaystyle\text{$\begin{gathered} P(m, n) = S(m,n) - Q(m,n)\end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered} = (4mn - m^{2}+ 3n^{2} + 4m^{2}n^{2}) - (5m^{2} - 8mn + 9n^{2} - 6m^{2}n^{2})\end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered} = 4mn - m^{2} + 3n^{2} + 4m^{2}n^{2} - 5m^{2} + 8mn - 9n^{2} + 6m^{2}n^{2}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = 12mn - 6m^{2} - 6n^{2} + 10m^{2}n^{2}\end{gathered}$}[/tex]
Portanto, o polinômio é:
[tex]\Large\displaystyle\text{$\begin{gathered} P(m, n) = 12mn - 6m^{2} - 6n^{2} + 10m^{2}n^{2}\end{gathered}$}[/tex]
[tex]\LARGE\displaystyle\text{$\begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered}$}[/tex]
Saiba mais: