One.
And you don’t need to calculate them to know that.
First, you know that polynomials of grade three have either one or three real roots. Second, you know that to have three real roots, they have to have a local maximum and a local minimum.
Take the derivative of your polynomial
()′()=23+8−7=62+8 f(x) =2
x
3
+8x−7
f
′
(x) =6
x
2
+8
Find the possible zeros of the derivative
′()=62+8=0
f
′
(
x
)
=
6
x
2
+
8
=
0
and see that there are no real zeros of the derivative. So the polynomial has no local maxima or minima, so it has exactly one real zero.
For the same reason every polynomial of the form
3++
a
x
3
+
b
x
+
c
with ,>0
a
,
b
>
0
has exactly one real root
15
2
