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# 12.5E: Ejercicios

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### La práctica hace a la perfección

##### Ejercicio $$\PageIndex{19}$$ Usa el Triángulo de Pascal para Expandir un Binomial

En los siguientes ejercicios, expande cada binomio usando el Triángulo de Pascal.

1. $$(x+y)^{4}$$
2. $$(a+b)^{8}$$
3. $$(m+n)^{10}$$
4. $$(p+q)^{9}$$
5. $$(x-y)^{5}$$
6. $$(a-b)^{6}$$
7. $$(x+4)^{4}$$
8. $$(x+5)^{3}$$
9. $$(y+2)^{5}$$
10. $$(y+1)^{7}$$
11. $$(z-3)^{5}$$
12. $$(z-2)^{6}$$
13. $$(4x-1)^{3}$$
14. $$(3x-1)^{5}$$
15. $$(3 x-4)^{4}$$
16. $$(3 x-5)^{3}$$
17. $$(2 x+3 y)^{3}$$
18. $$(3 x+5 y)^{3}$$
Responder

2. $$\begin{array}{l}{a^{8}+8 a^{7} b+28 a^{6} b^{2}+56 a^{5} b^{3}} {+70 a^{4} b^{4}+56 a^{3} b^{5}+28 a^{2} b^{6}} {+8 a b^{7}+b^{8}}\end{array}$$

4. $$\begin{array}{l}{p^{9}+9 p^{8} q+36 p^{7} q^{2}+84 p^{6} q^{3}} {+126 p^{5} q^{4}+126 p^{4} q^{5}+84 p^{3} q^{6}} {+36 p^{2} q^{7}+9 p q^{8}+q^{9}}\end{array}$$

6. $$\begin{array}{l}{a^{6}-6 a^{5} b+15 a^{4} b^{2}-20 a^{3} b^{3}} {+15 a^{2} b^{4}-6 a b^{5}+b^{6}}\end{array}$$

8. $$x^{3}+15 x^{2}+75 x+125$$

10. $$\begin{array}{l}{y^{7}+7 y^{6}+21 y^{5}+35 y^{4}+35 y^{3}} {+21 y^{2}+7 y+1}\end{array}$$

12. $$\begin{array}{l}{z^{6}-12 z^{5}+60 z^{4}-160 z^{3}+240 z^{2}} \\ {-192 z+64}\end{array}$$

14. $$\begin{array}{l}{243 x^{5}-405 x^{4}+270 x^{3}-90 x^{2}} {+15 x-1}\end{array}$$

16. $$27 x^{3}-135 x^{2}+225 x-125$$

18. $$27 x^{3}+135 x^{2} y+225 x y^{2}+125 y^{3}$$

##### Ejercicio $$\PageIndex{20}$$ Evaluar un Coeficiente Binomial
1. $$\left( \begin{array}{l}{8} \\ {1}\end{array}\right)$$
2. $$\left( \begin{array}{l}{10} \\ {10}\end{array}\right)$$
3. $$\left( \begin{array}{l}{6} \\ {0}\end{array}\right)$$
4. $$\left( \begin{array}{l}{9} \\ {3}\end{array}\right)$$
1. $$\left( \begin{array}{l}{7} \\ {1}\end{array}\right)$$
2. $$\left( \begin{array}{l}{4} \\ {4}\end{array}\right)$$
3. $$\left( \begin{array}{l}{3} \\ {0}\end{array}\right)$$
4. $$\left( \begin{array}{l}{5} \\ {3}\end{array}\right)$$
1. $$\left( \begin{array}{l}{3} \\ {1}\end{array}\right)$$
2. $$\left( \begin{array}{l}{9} \\ {9}\end{array}\right)$$
3. $$\left( \begin{array}{l}{7} \\ {0}\end{array}\right)$$
4. $$\left( \begin{array}{l}{5} \\ {3}\end{array}\right)$$
1. $$\left( \begin{array}{l}{4} \\ {1}\end{array}\right)$$
2. $$\left( \begin{array}{l}{5} \\ {5}\end{array}\right)$$
3. $$\left( \begin{array}{l}{8} \\ {0}\end{array}\right)$$
4. $$\left( \begin{array}{l}{11} \\ {9}\end{array}\right)$$
Responder

2.

1. $$7$$
2. $$1$$
3. $$1$$
4. $$45$$

4.

1. $$4$$
2. $$1$$
3. $$1$$
4. $$55$$
##### Ejercicio $$\PageIndex{21}$$ Utilizar el Teorema Binomial para Expandir un Binomial

En los siguientes ejercicios, expande cada binomio.

1. $$(x+y)^{3}$$
2. $$(m+n)^{5}$$
3. $$(a+b)^{6}$$
4. $$(s+t)^{7}$$
5. $$(x-2)^{4}$$
6. $$(y-3)^{4}$$
7. $$(p-1)^{5}$$
8. $$(q-4)^{3}$$
9. $$(3x-y)^{5}$$
10. $$(5x-2y)^{4}$$
11. $$(2x+5y)^{4}$$
12. $$(3x+4y)^{5}$$
Responder

2. $$\begin{array}{l}{m^{5}+5 m^{4} n+10 m^{3} n^{2}+10 m^{2} n^{3}} {+5 m n^{4}+n^{5}}\end{array}$$

4. $$\begin{array}{l}{s^{7}+7 s^{6} t+21 s^{5} t^{2}+35 s^{4} t^{3}} {+35 s^{3} t^{4}+21 s^{2} t^{5}+7 s t^{6}+t^{7}}\end{array}$$

6. $$y^{4}-12 y^{3}+54 y^{2}-108 y+81$$

8. $$q^{3}-12 q^{2}+48 q-64$$

10. $$\begin{array}{l}{625 x^{4}-1000 x^{3} y+600 x^{2} y^{2}} {-160 x y^{3}+16 y^{4}}\end{array}$$

12. $$\begin{array}{l}{243 x^{5}+1620 x^{4} y+4320 x^{3} y^{2}} {+5760 x^{2} y^{3}+3840 x y^{4}+1024 y^{5}}\end{array}$$

##### Ejercicio $$\PageIndex{22}$$ Utilizar el Teorema Binomial para Expandir un Binomial

En los siguientes ejercicios, encuentra el término indicado en la expansión del binomio.

1. Sexto periodo de $$(x+y)^{10}$$
2. Quinto periodo de $$(a+b)^{9}$$
3. Cuarto mandato de $$(x-y)^{8}$$
4. Séptimo periodo de $$(x-y)^{11}$$
Responder

2. $$126a^{5} b^{4}$$

4. $$462x^{5} y^{6}$$

##### Ejercicio $$\PageIndex{23}$$ Utilizar el Teorema Binomial para Expandir un Binomial

En los siguientes ejercicios, encontrar el coeficiente del término indicado en la expansión del binomio.

1. $$y^{3}$$ plazo de $$(y+5)^{4}$$
2. $$x^{6}$$ plazo de $$(x+2)^{8}$$
3. $$x^{5}$$ plazo de $$(x-4)^{6}$$
4. $$x^{7}$$ plazo de $$(x-3)^{9}$$
5. $$a^{4} b^{2}$$ plazo de $$(2 a+b)^{6}$$
6. $$p^{5} q^{4}$$ plazo de $$(3 p+q)^{9}$$
Responder

2. $$112$$

4. $$324$$

6. $$30,618$$

##### Ejercicios de $$\PageIndex{24}$$ escritura de ejercicios
1. En tus propias palabras explica cómo encontrar las filas del Triángulo de Pascal. Escribe las primeras cinco filas del Triángulo de Pascal.
2. En sus propias palabras, explique el patrón de exponentes para cada variable en la expansión de.
3. En tus propias palabras, explica la diferencia entre $$(a+b)^{n}$$ y $$(a-b)^{n}$$.
4. En tus propias palabras, explica cómo encontrar un término específico en la expansión de un binomio sin ampliar todo el asunto. Usa un ejemplo para ayudar a explicar.
Responder

2. Las respuestas variarán

4. Las respuestas variarán

## Autocomprobación

a. Después de completar los ejercicios, utilice esta lista de verificación para evaluar su dominio de los objetivos de esta sección.

b. En una escala de 1-10, ¿cómo calificaría su dominio de esta sección a la luz de sus respuestas en la lista de verificación? ¿Cómo se puede mejorar esto?

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