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Determine el coeficiente del monomio \[ \mathrm{T}(x, y)=2^{m} \cdot 5(m+n) x^{m-n} y^{2 m+n} \] si es de noveno grado y el \( \operatorname{GR}(y)=8 \)

Ask by Rose Martin. in Peru
Mar 17,2025

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El coeficiente del monomio es 200.

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Para determinar el coeficiente del monomio \( \mathrm{T}(x, y) = 2^{m} \cdot 5(m+n) x^{m-n} y^{2m+n} \) que sea de noveno grado, primero debemos analizar las condiciones dadas. La suma de los grados de \( x \) y \( y \) debe ser igual a 9, es decir: \[ (m - n) + (2m + n) = 9. \] Esto simplifica a: \[ 3m = 9 \Rightarrow m = 3. \] Luego, usando la condición \( \operatorname{GR}(y) = 8 \), se tiene que: \[ 2m + n = 8. \] Sustituyendo \( m = 3 \): \[ 2(3) + n = 8 \Rightarrow 6 + n = 8 \Rightarrow n = 2. \] Con \( m = 3 \) y \( n = 2 \), ahora sustituimos estos valores en la expresión del monomio para encontrar el coeficiente: \[ \mathrm{T}(x, y) = 2^{3} \cdot 5(3+2) x^{3-2} y^{2 \cdot 3 + 2} = 2^{3} \cdot 5(5) x^{1} y^{8}. \] Calculamos \( 2^{3} = 8 \) y \( 5 \cdot 5 = 25 \): \[ \mathrm{T}(x, y) = 8 \cdot 25 x y^{8} = 200 x y^{8}. \] Por lo tanto, el coeficiente del monomio es \( 200 \).

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