If 
, for how many real numbers 
does 
?
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One
Two
Three
Four
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The correct answer is E.
To determine how many real numbers satisfy 
is to determine how many solutions the equation 
has, which in turn is to determine how many solutions the equation 
has. It is not difficult to see, using the Rational Roots Theorem, that 
has no factor of the form 
for any whole or rational number value of , but there are real number values of such that is a factor. This problem must be solved using a non-algebraic method.
One way to determine how many numbers satisfy is to examine the graph of the function 
. Use a graphing calculator to graph 
for 
on a suitably large interval to see all intersections of the graph with the line 
, and then count the number of points of intersection.
There are at least four such points: A first point with -coordinate between 
and 
, a second point with -coordinate between and 
, a third point with -coordinate between and 
and a fourth point with -coordinate between 
and 
. The fact that 
is a polynomial of degree means that there can be at most four such points. Therefore, there are four values of for which .
Alternatively, one can examine a table of values of the function 
and then identify intervals for which the values of 
at the endpoints have different signs and apply the Intermediate Value Theorem to each of those intervals. Create a table of values for 
for whole number values of between
and , inclusive, and count the number of intervals of length for which the value of 
is greater than 
for one of the endpoints and less than for the other endpoint.
There are four such intervals: 
, 
, 
and 
. By the Intermediate Value Theorem, for each of these four intervals there is a value in that interval such that . This shows that there are at least four such values of 
and the fact that is a polynomial of degree means that there can be at most four such values of 
Therefore, there are four values of for which .