Prove that there is at least one real solution to the equation... $x^{17}+\frac{243}{1+x^4}=120$
Can anyone show me how to approach this problem..? Any help would be great, thanks.
 A: The left side is less than $120$ at, for example, $x=-2$, and bigger than $120$ at $x=-1$, so by the Intermediate Value Theorem it is exactly $120$ somewhere in between $-2$ and $-1$.  
We are using the fact that the function $x^{17}+\frac{243}{1+x^4}$ is continuous everywhere. 
A: HINT Multiply both sides by $1+x^4$. Then your equation simplifies to a $21$-degree polynomial. Any polynomial of odd degree satisfies which property?
A: Because $17$ is odd, $x^{17}$ goes to $\infty$ as $x$ goes to $\infty$ , and $x^{17}$ goes to $-\infty$  as $x$ goes to $-\infty$. 
On the other hand ${243 \over 1 + x^4}$ goes to zero as $x$ goes to $\infty$ and $-\infty$, so it stays bounded. 
Hence $x^{17} + {243 \over 1 + x^4}$ goes to $\infty$ as $x$ goes to $\infty$, and goes to $-\infty$ as $x$ goes to $-\infty$. 
Since $x^{17} + {243 \over 1 + x^4}$ is a continuous function, the last statement implies that $x^{17} + {243 \over 1 + x^4}$ will take every real value, including $120$. The intermediate value theorem can be used to make this rigorous. 
