# Question #60f48

##### 1 Answer

The easiest way of doing this is using a calculus concept called the intermediate value theorem. The intermediate value theorem basically states that given a function

While a function being *continuous* has a more rigorous and nuanced definition, a non-calculus student may think of it as being a function which can be graphed without lifting one's pen off of the paper. This suffices for most simple functions, and also gives some idea as to why the intermediate value theorem works. If you never take your pen off the paper, then to get from

As it happens, a polynomial function will be continuous on any interval, and we can use that to show that the given equation has a solution. Consider, for example, the interval

Letting

and

As

In particular, as

If we look at the graph, the argument boils down to noting that as the function crosses the line

graph{x^9+3x+15 [-51.9, 52.1, -22.96, 29]}