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Para realizar la división del polinomio [tex]\(\left(-4x^3 y + 8x^2 y^2 + 12xy^3 \right)\)[/tex] entre [tex]\(4xy\)[/tex], procederemos a dividir cada término del numerador individualmente entre el divisor [tex]\(4xy\)[/tex]:
1. Primer término:
[tex]\[-4x^3 y \div 4xy\][/tex]
Simplificamos los coeficientes:
[tex]\[ -4 \div 4 = -1 \][/tex]
Simplificamos [tex]\(x^3 \div x\)[/tex]:
[tex]\[ x^{3-1} = x^2 \][/tex]
Simplificamos [tex]\(y \div y\)[/tex]:
[tex]\[ y^{1-1} = y^0 = 1 \][/tex]
Entonces, el primer término se simplifica a:
[tex]\[ \frac{-4x^3 y}{4xy} = -x^2 \][/tex]
2. Segundo término:
[tex]\[8x^2 y^2 \div 4xy\][/tex]
Simplificamos los coeficientes:
[tex]\[ 8 \div 4 = 2 \][/tex]
Simplificamos [tex]\(x^2 \div x\)[/tex]:
[tex]\[ x^{2-1} = x \][/tex]
Simplificamos [tex]\(y^2 \div y\)[/tex]:
[tex]\[ y^{2-1} = y \][/tex]
Entonces, el segundo término se simplifica a:
[tex]\[ \frac{8x^2 y^2}{4xy} = 2xy \][/tex]
3. Tercer término:
[tex]\[12xy^3 \div 4xy\][/tex]
Simplificamos los coeficientes:
[tex]\[ 12 \div 4 = 3 \][/tex]
Simplificamos [tex]\(x \div x\)[/tex]:
[tex]\[ x^{1-1} = x^0 = 1 \][/tex]
Simplificamos [tex]\(y^3 \div y\)[/tex]:
[tex]\[ y^{3-1} = y^2 \][/tex]
Entonces, el tercer término se simplifica a:
[tex]\[ \frac{12xy^3}{4xy} = 3y^2 \][/tex]
Finalmente, sumamos los términos simplificados:
[tex]\[ -x^2 + 2xy + 3y^2 \][/tex]
Entonces, la solución a la división [tex]\(\left(-4x^3 y + 8x^2 y^2 + 12xy^3 \right) \div 4xy\)[/tex] es:
[tex]\[ -x^2 + 2xy + 3y^2 \][/tex]
1. Primer término:
[tex]\[-4x^3 y \div 4xy\][/tex]
Simplificamos los coeficientes:
[tex]\[ -4 \div 4 = -1 \][/tex]
Simplificamos [tex]\(x^3 \div x\)[/tex]:
[tex]\[ x^{3-1} = x^2 \][/tex]
Simplificamos [tex]\(y \div y\)[/tex]:
[tex]\[ y^{1-1} = y^0 = 1 \][/tex]
Entonces, el primer término se simplifica a:
[tex]\[ \frac{-4x^3 y}{4xy} = -x^2 \][/tex]
2. Segundo término:
[tex]\[8x^2 y^2 \div 4xy\][/tex]
Simplificamos los coeficientes:
[tex]\[ 8 \div 4 = 2 \][/tex]
Simplificamos [tex]\(x^2 \div x\)[/tex]:
[tex]\[ x^{2-1} = x \][/tex]
Simplificamos [tex]\(y^2 \div y\)[/tex]:
[tex]\[ y^{2-1} = y \][/tex]
Entonces, el segundo término se simplifica a:
[tex]\[ \frac{8x^2 y^2}{4xy} = 2xy \][/tex]
3. Tercer término:
[tex]\[12xy^3 \div 4xy\][/tex]
Simplificamos los coeficientes:
[tex]\[ 12 \div 4 = 3 \][/tex]
Simplificamos [tex]\(x \div x\)[/tex]:
[tex]\[ x^{1-1} = x^0 = 1 \][/tex]
Simplificamos [tex]\(y^3 \div y\)[/tex]:
[tex]\[ y^{3-1} = y^2 \][/tex]
Entonces, el tercer término se simplifica a:
[tex]\[ \frac{12xy^3}{4xy} = 3y^2 \][/tex]
Finalmente, sumamos los términos simplificados:
[tex]\[ -x^2 + 2xy + 3y^2 \][/tex]
Entonces, la solución a la división [tex]\(\left(-4x^3 y + 8x^2 y^2 + 12xy^3 \right) \div 4xy\)[/tex] es:
[tex]\[ -x^2 + 2xy + 3y^2 \][/tex]
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