If , for how many real numbers
does
?
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None
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
.