Answer
The statement is false.
Two ways of making it true is changing
" ... gives the exact number ..."
to
"... gives the maximum number ..."
or "gives an estimate "
Work Step by Step
Descartes's Rule of Signs (page 384)
Let $f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{2}x^{2}+a_{1}x+a_{0}$
be a polynomial with real coefficients.
1. The number of positive real zeros of $f$ is either
$\mathrm{a}$. the same as the number of sign changes of $f(x)$
or
$\mathrm{b}$. less than the number of sign changes of $f(x)$ by a positive even integer.
If $f(x)$ has only one variation in sign, then $f$ has exactly one positive real zero.
2. The number of NEGATIVE real zeros of $f$ is either
$\mathrm{a}$. the same as the number of sign changes of $f(-x)$
or
$\mathrm{b}$. less than the number of sign changes of $f(-x)$ by a positive even integer.
If $f(-x)$ has only one variation in sign, then $f$ has exactly one negative real zero.
-----------------
So, Descartes's Rule generally gives ESTIMATES,
and the MAXIMUM number of positive/negative real roots.
It gives an exact number (one) only when the number of sign variations in f(x) or f(-x) is 1
The statement is false.
Two ways of making it true is changing
" ... gives the exact number ..."
to
"... gives the maximum number ..."
or "gives an estimate "
