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a) \( 25 x^{2} y^{22}-16 m^{10} n^{4} p^{8} \) b) \( -121 a^{6} b^{12} c^{24}+81 d^{32} e^{44} f^{28} \) c) \( \frac{36}{81} x^{4} z^{18}-\frac{100}{169} y^{36} w^{12} \) d) \( -\frac{49}{25} a^{6} b^{14}+\frac{1}{9} c^{26} d^{30} \)

Ask by Alexander Sandoval. in Colombia
Nov 07,2024

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Answer

a) \( (5y^{11} - 4m^{5} n^{2} p^{4})(5y^{11} + 4m^{5} n^{2} p^{4}) \) b) \( (9 d^{16} e^{22} f^{14} - 11 a^{3} b^{6} c^{12})(9 d^{16} e^{22} f^{14} + 11 a^{3} b^{6} c^{12}) \) c) \( \left(\frac{2}{3} x^{2} z^{9} - \frac{10}{13} y^{18} w^{6}\right)\left(\frac{2}{3} x^{2} z^{9} + \frac{10}{13} y^{18} w^{6}\right) \) d) \( \left(\frac{1}{3} c^{13} d^{15} - \frac{7}{5} a^{3} b^{7}\right)\left(\frac{1}{3} c^{13} d^{15} + \frac{7}{5} a^{3} b^{7}\right) \)

Solution

Para factorizar las expresiones dadas, procederemos con cada una de ellas: a) \( 25 x^{2} y^{22}-16 m^{10} n^{4} p^{8} \) Esta expresión es una diferencia de cuadrados. Podemos reescribirla como: \[ (5y^{11})^{2} - (4m^{5} n^{2} p^{4})^{2} \] Por lo tanto, se puede factorizar como: \[ (5y^{11} - 4m^{5} n^{2} p^{4})(5y^{11} + 4m^{5} n^{2} p^{4}) \] b) \( -121 a^{6} b^{12} c^{24}+81 d^{32} e^{44} f^{28} \) Esta expresión también es una diferencia de cuadrados. Reescribiéndola, tenemos: \[ -(11 a^{3} b^{6} c^{12})^{2} + (9 d^{16} e^{22} f^{14})^{2} \] Por lo tanto, se puede factorizar como: \[ (9 d^{16} e^{22} f^{14} - 11 a^{3} b^{6} c^{12})(9 d^{16} e^{22} f^{14} + 11 a^{3} b^{6} c^{12}) \] c) \( \frac{36}{81} x^{4} z^{18}-\frac{100}{169} y^{36} w^{12} \) Primero simplificamos los coeficientes: \[ \frac{36}{81} = \frac{4}{9} \quad \text{y} \quad \frac{100}{169} = \frac{100}{169} \] La expresión se convierte en: \[ \frac{4}{9} x^{4} z^{18} - \frac{100}{169} y^{36} w^{12} \] Ahora, podemos reescribirla como una diferencia de cuadrados: \[ \left(\frac{2}{3} x^{2} z^{9}\right)^{2} - \left(\frac{10}{13} y^{18} w^{6}\right)^{2} \] Por lo tanto, se puede factorizar como: \[ \left(\frac{2}{3} x^{2} z^{9} - \frac{10}{13} y^{18} w^{6}\right)\left(\frac{2}{3} x^{2} z^{9} + \frac{10}{13} y^{18} w^{6}\right) \] d) \( -\frac{49}{25} a^{6} b^{14}+\frac{1}{9} c^{26} d^{30} \) Primero, reescribimos la expresión: \[ -\left(\frac{7}{5} a^{3} b^{7}\right)^{2} + \left(\frac{1}{3} c^{13} d^{15}\right)^{2} \] Esto también es una diferencia de cuadrados, por lo que se puede factorizar como: \[ \left(\frac{1}{3} c^{13} d^{15} - \frac{7}{5} a^{3} b^{7}\right)\left(\frac{1}{3} c^{13} d^{15} + \frac{7}{5} a^{3} b^{7}\right) \] Estas son las factorizaciones de las expresiones dadas.

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Extra Insights

If you take a closer look at the expressions you've shared, you'll notice that they can all be simplified using techniques from algebraic identities, particularly those involving differences of squares and greatest common factors. This is key because manipulation of terms can lead you to factor out common elements, therefore making simplification easier. When dealing with polynomial expressions, don't rush the process! A common mistake is forgetting to check for signs when combining like terms or overlooking factors that could have been simplified early on. Always double-check each step, and try to write the entire factorization clearly to avoid confusion—especially in multi-variable polynomials!

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