A cubic polynomials has 6 zeroes
Answers
We have seen that cubic polynomials are of the form
p
(
x
)
:
a
x
3
+
b
x
2
+
c
x
+
d
,
a
≠
0
Any such polynomial will have, in general, three zeroes. For example,
p
(
x
)
:
x
3
−
6
x
2
+
11
x
−
6
has the following three zeroes (verify that these are indeed the zeroes of the polynomial):
x
=
1
,
2
,
3
.
The three zeroes of a cubic polynomial might all be equal. For example, consider
p
(
x
)
:
(
x
−
1
)
3
. This has the three zeroes
x
=
1
,
1
,
1
,
which happen to be identical.
Another case which is possible is that two of the zeroes are equal, and the third is different. For example, consider
p
(
x
)
:
(
x
−
1
)
2
(
x
−
2
)
. This has the three zeroes:
x
=
1
,
1
,
2
.
Will a cubic polynomial always have three real zeroes? The answer is no. Just as a quadratic polynomial does not always have real zeroes, a cubic polynomial may also not have all its zeroes as real. But there is a crucial difference. A cubic polynomial will always have at least one real zero. Thus, the following cases are possible for the zeroes of a cubic polynomial:
Xd