Rojas Hanson
08/23/2024 · Junior High School
18. La solución para el siguiente binomio \( (a-3)^{3} \), es Aplique la formula para resolver el binoni A) \( a^{3}-9 a^{2}+9 a-27 \) B) \( a^{3}+9 a^{2}+27 a+27 \) C) \( a^{3}-9 a^{2}-27 a-27 \) D) \( a^{3}-9 a^{2}+27 a-27 \)
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Step-by-step Solution
Para resolver el binomio \( (a-3)^{3} \), podemos utilizar la fórmula del cubo de un binomio, que es:
\[
(x - y)^{3} = x^{3} - 3x^{2}y + 3xy^{2} - y^{3}
\]
En este caso, \( x = a \) y \( y = 3 \). Sustituyendo en la fórmula, tenemos:
\[
(a - 3)^{3} = a^{3} - 3(a^{2})(3) + 3(a)(3^{2}) - 3^{3}
\]
Ahora, calculamos cada término:
1. \( a^{3} \)
2. \( -3(a^{2})(3) = -9a^{2} \)
3. \( 3(a)(3^{2}) = 3(a)(9) = 27a \)
4. \( -3^{3} = -27 \)
Ahora, juntamos todos los términos:
\[
(a - 3)^{3} = a^{3} - 9a^{2} + 27a - 27
\]
Por lo tanto, la solución para el binomio \( (a-3)^{3} \) es:
\[
a^{3} - 9a^{2} + 27a - 27
\]
La respuesta correcta es la opción **D**: \( a^{3} - 9a^{2} + 27a - 27 \).
Quick Answer
La respuesta correcta es D: \( a^{3} - 9a^{2} + 27a - 27 \).
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