Question

Expand (2+x)^-3







....

Answer 1

Hello,

Explanation:

`(a+x)^n=a^n+\left(\begin{array}{c}n\\ 1\end{array}\right)*a^(n-1)*x+\left(\begin{array}{c}n\\ 2\end{array}\right)*a^(n-2)*x^2+\left(\begin{array}{c}n\\ 3\end{array}\right)*a^(n-3)*x^3+\left(\begin{array}{c}n\\ 4\end{array}\right)*a^(n-4)*x^4+...+\left(\begin{array}{c}n\\ n\end{array}\right)*a^(n-n)*x^n`

`with \\\\\left(\begin{array}{c}n\\ 1\end{array}\right)=n\\\\\left(\begin{array}{c}n\\ 2\end{array}\right)=(n(n-1))/(2!) \\\\\left(\begin{array}{c}n\\3 \end{array}\right)=(n(n-1)(n-2))/(3!) \\\\...\\`

`(1)/((2+x)^3) =(1)/(8) +3*(x)/(4)+3(x^2)/(2)+x^3\\\\`

Answer 2

1/(x^3 + 6x^2 + 12x + 8)

Explanation:

The first thing we do is rationalize this expression. (2+x)^-3 is written as

1/(2+x)^3

Then from there we can foil out the denominator. It is easiest to foil (2+x)(2+x) first and then multiply that product by (2+x).

(2+x)(2+x) = 4 + 4x + x^2

(4+4x+x^2)(2+x) = 8+8x+2x^2+4x+4x^2+x^3.

Then we combine like terms and put them in order to get:

x^3 + 6x^2 + 12x + 8

And of course we can't forget that this was raised to the negative third power, so our answer is 1/(x^3 + 6x^2 + 12x + 8)