Consider the function $f(x)=\frac{x^2(x+1)^3}{(x-2)^2(x-4)^4}$

Question

$$f(x)=\frac{x^2(x+1)^3}{(x-2)^2(x-4)^4}$$

The text tells me to divide by $x^6$ because it wants me to graph but I would like some guidance on rewriting the equation

So now:

$$\Large f(x)=\frac{\frac{x^2}{x^3}\cdot\frac{(x+1)^3}{x^3}}{\frac{(x-2)^2}{x^2}\cdot\frac{(x-4)^4}{x^4}}=\frac{\frac{1}{x}\left( 1+\frac{1}{x}\right)^3}{\left(1-\frac{2}{x}\right)^2\left(1-\frac{4}{x}\right)^4}$$

My question is why does this: $$\frac{(x+1)^3}{x^3}$$ equal this: $$\left(1+\frac{1}{x}\right)^3$$

Answer 1

We have

$$\frac{(x+1)^3}{x^3}=\left(\frac{x+1}{x}\right)^3=\left(\frac x x+\frac1x\right)^3=\left(1+\frac1x\right)^3$$

Answer 2

That is after you bring it to a common exponent $3 $ in $\ \,(x+1/x)^3 $