Fowler Luna
04/28/2024 · High School
6. PPeraciones multiplicalivas: entie polinom Realiza las siguientes oleraciones \( (x+3)(x+5) \) \( (2 x-5)(x+4) \) \( \left(\frac{6}{5} x^{2}-2 x y+\frac{7}{4} y^{2}\right)\left(2 x^{2}+3 y^{2}\right) \)
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Vamos a realizar las multiplicaciones de los polinomios que has proporcionado.
1. **Multiplicación de \( (x+3)(x+5) \)**:
\[
(x+3)(x+5) = x^2 + 5x + 3x + 15 = x^2 + 8x + 15
\]
2. **Multiplicación de \( (2x-5)(x+4) \)**:
\[
(2x-5)(x+4) = 2x \cdot x + 2x \cdot 4 - 5 \cdot x - 5 \cdot 4
\]
\[
= 2x^2 + 8x - 5x - 20 = 2x^2 + 3x - 20
\]
3. **Multiplicación de \( \left(\frac{6}{5} x^{2}-2 x y+\frac{7}{4} y^{2}\right)\left(2 x^{2}+3 y^{2}\right) \)**:
Usamos la propiedad distributiva (también conocida como el método FOIL para binomios):
\[
\left(\frac{6}{5} x^{2}\right)(2x^{2}) + \left(\frac{6}{5} x^{2}\right)(3y^{2}) + \left(-2xy\right)(2x^{2}) + \left(-2xy\right)(3y^{2}) + \left(\frac{7}{4}y^{2}\right)(2x^{2}) + \left(\frac{7}{4}y^{2}\right)(3y^{2})
\]
Calculamos cada término:
1. \( \frac{6}{5} x^{2} \cdot 2x^{2} = \frac{12}{5} x^{4} \)
2. \( \frac{6}{5} x^{2} \cdot 3y^{2} = \frac{18}{5} x^{2}y^{2} \)
3. \( -2xy \cdot 2x^{2} = -4x^{3}y \)
4. \( -2xy \cdot 3y^{2} = -6xy^{3} \)
5. \( \frac{7}{4}y^{2} \cdot 2x^{2} = \frac{7}{2} x^{2}y^{2} \)
6. \( \frac{7}{4}y^{2} \cdot 3y^{2} = \frac{21}{4} y^{4} \)
Ahora, sumamos todos los términos:
\[
\frac{12}{5} x^{4} + \left(\frac{18}{5} + \frac{7}{2}\right)x^{2}y^{2} - 4x^{3}y - 6xy^{3} + \frac{21}{4} y^{4}
\]
Para combinar \( \frac{18}{5} + \frac{7}{2} \), encontramos un común denominador (10):
\[
\frac{18}{5} = \frac{36}{10}, \quad \frac{7}{2} = \frac{35}{10} \quad \Rightarrow \quad \frac{36}{10} + \frac{35}{10} = \frac{71}{10}
\]
Por lo tanto, el resultado final es:
\[
\frac{12}{5} x^{4} - 4x^{3}y + \frac{71}{10} x^{2}y^{2} - 6xy^{3} + \frac{21}{4} y^{4}
\]
Así que los resultados de las multiplicaciones son:
1. \( x^2 + 8x + 15 \)
2. \( 2x^2 + 3x - 20 \)
3. \( \frac{12}{5} x^{4} - 4x^{3}y + \frac{71}{10} x^{2}y^{2} - 6xy^{3} + \frac{21}{4} y^{4} \)
Quick Answer
1. \( x^2 + 8x + 15 \)
2. \( 2x^2 + 3x - 20 \)
3. \( \frac{12}{5} x^{4} - 4x^{3}y + \frac{71}{10} x^{2}y^{2} - 6xy^{3} + \frac{21}{4} y^{4} \)
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