Question

1-Qual é o termo em x⁵ no desenvolvimento de (x+3)⁸ ?

2-Determine a soma dos coeficientes do desenvolvimento de (x-3y)⁷ ?

3-Qual a soma dos coeficientes dos termos do desenvolvimento de:

a-(2x-3y)¹²

b-(x-y)⁵⁰

4-Determine o 7° termo do binômio (2x+1)9 ?​

Answer 1

$\boxed{(a+b)^n=\sum\limits_{p=0}^{n}{\dbinom{n}{p}a^{n-p}b^p}}$

$\boxed{T_{p+1}=\dbinom{n}{p}a^{n-p}b^p}$

$\boxed{\dbinom{n}{p}=\dfrac{n!}{p!(n-p)!}}$

$\boxed{\text{Soma dos coeficientes de}\ (ra+sb)^n=(r+s)^n}$

1.

$(x+3)^8\ \to\ a=x,\ b=3,\ n=8$

$T_{p+1}=\dbinom{8}{p}x^{8-p}3^p$

$x^5=x^{8-p}\ \to\ 5=8-p\ \to\ \boxed{p=3}$

$T_{3+1}=\dbinom{8}{3}x^{8-3}3^3=(56)x^5(27)\ \to\ \boxed{T_4=1512x^5}$

2.

$\text{Soma dos coeficientes de}\ (x-3y)^7=(1-3)^7$

$(1-3)^7=(-2)^7=\boxed{-128}$

3.

a)

$\text{Soma dos coeficientes de}\ (2x-3y)^{12}=(2-3)^{12}$

$(2-3)^{12}=(-1)^{12}=\boxed{1}$

b)

$\text{Soma dos coeficientes de}\ (x-y)^{50}=(1-1)^{50}$

$(1-1)^{50}=0^{50}=\boxed{0}$

4.

$(2x+1)^9\ \to\ a=2x,\ b=1,\ n=9$

$T_{p+1}=\dbinom{9}{p}(2x)^{9-p}1^p\ \to\ T_{p+1}=\dbinom{9}{p}(2x)^{9-p}$

$p+1=7\ \to\ \boxed{p=6}$

$T_{6+1}=\dbinom{9}{6}(2x)^{9-6}=(84)(2x)^3=(84)(8)x^3\ \to\ \boxed{T_7=672x^3}$