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Popular Álgebra >

simplificar l^3(m-n)^3+m^3(n-l)^3+n^3*(l-m)^3

Solução

simplificar

l^{3} \cdot (m - n)^{3} + m^{3} \cdot (n - l)^{3} + n^{3} \cdot (l - m)^{3}

Solução

3 l^{3} m n^{2} - 3 l^{3} m^{2} n - 3 l^{2} m n^{3} + 3 l^{2} m^{3} n + 3 l m^{2} n^{3} - 3 l m^{3} n^{2}

Passos da solução

l^{3} (m - n)^{3} + m^{3} (n - l)^{3} + n^{3} (l - m)^{3}

(m - n)^{3} : m^{3} - 3 m^{2} n + 3 m n^{2} - n^{3}

= l^{3} (m^{3} - 3 m^{2} n + 3 m n^{2} - n^{3}) + m^{3} (n - l)^{3} + n^{3} (l - m)^{3}

(n - l)^{3} : n^{3} - 3 n^{2} l + 3 n l^{2} - l^{3}

= l^{3} (m^{3} - 3 m^{2} n + 3 m n^{2} - n^{3}) + m^{3} (n^{3} - 3 n^{2} l + 3 n l^{2} - l^{3}) + n^{3} (l - m)^{3}

(l - m)^{3} : l^{3} - 3 l^{2} m + 3 l m^{2} - m^{3}

= l^{3} (m^{3} - 3 m^{2} n + 3 m n^{2} - n^{3}) + m^{3} (n^{3} - 3 n^{2} l + 3 n l^{2} - l^{3}) + n^{3} (l^{3} - 3 l^{2} m + 3 l m^{2} - m^{3})

Expandir

l^{3} (m^{3} - 3 m^{2} n + 3 m n^{2} - n^{3}) : l^{3} m^{3} - 3 l^{3} m^{2} n + 3 l^{3} m n^{2} - l^{3} n^{3}

= l^{3} m^{3} - 3 l^{3} m^{2} n + 3 l^{3} m n^{2} - l^{3} n^{3} + m^{3} (n^{3} - 3 n^{2} l + 3 n l^{2} - l^{3}) + n^{3} (l^{3} - 3 l^{2} m + 3 l m^{2} - m^{3})

Expandir

m^{3} (n^{3} - 3 n^{2} l + 3 n l^{2} - l^{3}) : m^{3} n^{3} - 3 l m^{3} n^{2} + 3 l^{2} m^{3} n - l^{3} m^{3}

= l^{3} m^{3} - 3 l^{3} m^{2} n + 3 l^{3} m n^{2} - l^{3} n^{3} + m^{3} n^{3} - 3 l m^{3} n^{2} + 3 l^{2} m^{3} n - l^{3} m^{3} + n^{3} (l^{3} - 3 l^{2} m + 3 l m^{2} - m^{3})

Expandir

n^{3} (l^{3} - 3 l^{2} m + 3 l m^{2} - m^{3}) : l^{3} n^{3} - 3 l^{2} m n^{3} + 3 l m^{2} n^{3} - m^{3} n^{3}

= l^{3} m^{3} - 3 l^{3} m^{2} n + 3 l^{3} m n^{2} - l^{3} n^{3} + m^{3} n^{3} - 3 l m^{3} n^{2} + 3 l^{2} m^{3} n - l^{3} m^{3} + l^{3} n^{3} - 3 l^{2} m n^{3} + 3 l m^{2} n^{3} - m^{3} n^{3}

Simplificar

l^{3} m^{3} - 3 l^{3} m^{2} n + 3 l^{3} m n^{2} - l^{3} n^{3} + m^{3} n^{3} - 3 l m^{3} n^{2} + 3 l^{2} m^{3} n - l^{3} m^{3} + l^{3} n^{3} - 3 l^{2} m n^{3} + 3 l m^{2} n^{3} - m^{3} n^{3} : 3 l^{3} m n^{2} - 3 l^{3} m^{2} n - 3 l^{2} m n^{3} + 3 l^{2} m^{3} n + 3 l m^{2} n^{3} - 3 l m^{3} n^{2}

= 3 l^{3} m n^{2} - 3 l^{3} m^{2} n - 3 l^{2} m n^{3} + 3 l^{2} m^{3} n + 3 l m^{2} n^{3} - 3 l m^{3} n^{2}

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