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# 25.3: Ejercicios

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## Ejercicio$$\PageIndex{1}$$

Encontrar el valor del coeficiente factorial o binomial.

1. $$5!$$
2. $$3!$$
3. $$9!$$
4. $$2!$$
5. $$0!$$
6. $$1!$$
7. $$19!$$
8. $$64!$$
9. $$\dbinom{5}{2}$$
10. $$\dbinom{9}{6}$$
11. $$\dbinom{12}{1}$$
12. $$\dbinom{12}{0}$$
13. $$\dbinom{23}{22}$$
14. $$\dbinom{19}{12}$$
15. $$\dbinom{13}{11}$$
16. $$\dbinom{16}{5}$$
Contestar
1. $$120$$
2. $$6$$
3. $$362880$$
4. $$2$$
5. $$1$$
6. $$1$$
7. $$\approx 1.216 \cdot 10^{17}$$
8. $$\approx 1.269 \cdot 10^{89}$$
9. $$10$$
10. $$84$$
11. $$12$$
12. $$1$$
13. $$23$$
14. $$50388$$
15. $$78$$
16. $$4368$$

## Ejercicio$$\PageIndex{2}$$

Expandir la expresión a través del teorema binomial.

1. $$(m+n)^{4}$$
2. $$(x+2)^5$$
3. $$(x-y)^6$$
4. $$(-p-q)^5$$
Contestar
1. $$m^{4}+4 m^{3} n+6 m^{2} n^{2}+4 m n^{3}+n^{4}$$
2. $$x^{5}+10 x^{4}+40 x^{3}+80 x^{2}+80 x+32$$
3. $$x^{6}-6 x^{5} y+15 x^{4} y^{2}-20 x^{3} y^{3}+15 x^{2} y^{4}-6 x y^{5}+y^{6}$$
4. $$-p^{5}-5 p^{4} q-10 p^{3} q^{2}-10 p^{2} q^{3}-5 p q^{4}-q^{5}$$

## Ejercicio$$\PageIndex{3}$$

Expandir la expresión.

1. $$(x-2y)^3$$
2. $$(x-10)^4$$
3. $$(x^2y+y^2)^5$$
4. $$(2y^2-5x^4)^4$$
5. $$(x+\sqrt{x})^3$$
6. $$\left(-2\dfrac{x^2}{y}-\dfrac{y^3}{x}\right)^5$$
7. $$(\sqrt{2}-2\sqrt{3})^3$$
8. $$(1-i)^3$$
Contestar
1. $$x^{3}-6 x^{2} y+12 x y^{2}-8 y^{3}$$
2. $$x^{4}-40 x^{3}+600 x^{2}-4000 x+10000$$
3. $$x^{10} y^{5}+5 x^{8} y^{6}+10 x^{6} y^{7}+10 x^{4} y^{8}+5 x^{2} y^{9}+y^{10}$$
4. $$16 y^{8}-160 x^{4} y^{6}+600 x^{8} y^{4}-1000 x^{12} y^{2}+625 x^{16}$$
5. $$x^{3}+3 x^{\frac{5}{2}}+3 x^{2}+x^{\frac{3}{2}}$$
6. $$-32 \dfrac{x^{10}}{y^{5}}-80 \dfrac{x^{7}}{y}-80 x^{4} y^{3}-40 x y^{7}-10 \dfrac{y^{11}}{x^{2}}-\dfrac{y^{15}}{x^{5}}$$
7. $$38 \sqrt{2}-36 \sqrt{3}$$
8. $$-2-2 i$$

## Ejercicio$$\PageIndex{4}$$

Determinar:

1. los primeros$$3$$ términos en la expansión binomial de$$(xy-4x)^{5}$$
2. los primeros$$2$$ términos en la expansión binomial de$$(2a^2+b^3)^{9}$$
3. los últimos$$3$$ términos en la expansión binomial de$$(3y^2-x^2)^{7}$$
4. los primeros$$3$$ términos en la expansión binomial de$$\left(\dfrac{x}{y}-\dfrac{y}{x}\right)^{10}$$
5. los últimos$$4$$ términos en la expansión binomial de$$\left(m^3n+\dfrac{1}{2}n^2\right)^{6}$$
Contestar
1. $$x^{5} y^{5}-20 x^{5} y^{4}+160 x^{5} y^{3}$$
2. $$512 a^{18}+2304 a^{16} b^{3}$$
3. $$-189 x^{10} y^{4}+21 x^{12} y^{2}-x^{14}$$
4. $$\dfrac{x^{10}}{y^{10}}-10 \dfrac{x^{8}}{y^{8}}+45 \dfrac{x^{6}}{y^{6}}$$
5. $$\dfrac{5}{2} m^{9} n^{9}+\dfrac{15}{16} m^{6} n^{10}+\dfrac{3}{16} m^{3} n^{11}+\dfrac{1}{64} n^{12}$$

## Ejercicio$$\PageIndex{5}$$

Determinar:

1. el término$$5$$ th en la expansión binomial de$$(x+y)^{7}$$
2. el término$$3$$ rd en la expansión binomial de$$(x^2-y)^{9}$$
3. el término$$10$$ th en la expansión binomial de$$(2-w)^{11}$$
4. el término$$5$$ th en la expansión binomial de$$(2x+xy)^{7}$$
5. el término$$7$$ th en la expansión binomial de$$(2a+5b)^{6}$$
6. el término$$6$$ th en la expansión binomial de$$(3p^2-q^3p)^{7}$$
7. el término$$10$$ th en la expansión binomial de$$\left(4+\dfrac{b}{2}\right)^{13}$$
Contestar
1. $$35 x^{3} y^{4}$$
2. $$36 x^{14} y^{2}$$
3. $$-220 w^{9}$$
4. $$280 x^{7} y^{4}$$
5. $$15625 b^{6}$$
6. $$-189 p^{9} q^{15}$$
7. $$\dfrac{715}{2} b^{9}$$

## Ejercicio$$\PageIndex{6}$$

Determinar:

1. el$$x^3y^{6}$$ término en la expansión binomial de$$(x+y)^{9}$$
2. el$$r^4s^4$$ término en la expansión binomial de$$(r^2-s)^{6}$$
3. el$$x^{4}$$ término en la expansión binomial de$$(x-1)^{11}$$
4. el$$x^3y^{6}$$ término en la expansión binomial de$$(x^3+5y^2)^{4}$$
5. el$$x^{7}$$ término en la expansión binomial de$$(2x-x^2)^{5}$$
6. la parte imaginaria del número$$(1+i)^3$$
Contestar
1. $$84 x^{3} y^{6}$$
2. $$15 r^{4} s^{4}$$
3. $$-330 x^{4}$$
4. $$500 x^{3} y^{6}$$
5. $$80 x^{7}$$
6. $$2 i$$

This page titled 25.3: Ejercicios is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.