Answer
$f(1)$ and $f(2)$ have opposite signs,
so f(x) has a real zero between $1$ and $2$
Work Step by Step
The Intermediate Value Theorem for Polynomial Functions$:$
Let $f$ be a polynomial function with real coefficients.
If $f(a)$ and $f(b)$ have opposite signs,
then there is at least one value of $c$ between $a$ and $b$ for which $f(c)=0$.
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$ a=1,\quad f(a)=(1)^{5}-(1)^{3}-1=1-1-1=-1\quad$(negative)
$ b=2,\quad f(b)=(2)^{5}-(2)^{3}-1=32-8-1=23\quad$(positive)
$f(1)$ and $f(2)$ have opposite signs,
so f(x) has a real zero between $1$ and $2$
