if the sum of zeros of a cubic polynomial 2x cube minus kx square -8 x -12 minus 3 by 2 find the value of k with step by step explanation
Answers
Step-by-step explanation:
Find the values of '
k
' if equation
x
3
−
3
x
2
+
2
=
k
has:- (i)3 real roots (ii)1 real root?
Precalculus
1 Answer
George C.
Aug 11, 2018
(i) The given equation has
3
real roots when
k
∈
(
−
2
,
2
)
(ii) The given equation has
1
real root when
k
∈
(
−
∞
,
−
2
)
∪
(
2
,
∞
)
Explanation:
We could solve this with the aid of the cubic discriminant, but let's look at it without...
Given:
x
3
−
3
x
2
+
2
=
k
Let:
f
(
x
)
=
x
3
−
3
x
2
+
2
−
k
First note that if
k
=
0
then
x
=
1
is a real zero and
(
x
−
1
)
a factor:
x
3
−
3
x
2
+
2
=
(
x
−
1
)
(
x
2
−
2
x
−
2
)
x
3
−
3
x
2
+
2
=
(
x
−
1
)
(
x
2
−
2
x
+
1
−
3
)
x
3
−
3
x
2
+
2
=
(
x
−
1
)
(
(
x
−
1
)
2
−
(
√
3
)
2
)
x
3
−
3
x
2
+
2
=
(
x
−
1
)
(
x
−
1
−
√
3
)
(
x
−
1
+
√
3
)
which has
3
real zeros.
Next note that
f
(
x
)
will have a repeated zero if and only if it has a common factor with:
f
'
(
x
)
=
3
x
2
−
6
x
=
3
x
(
x
−
2
)
If
x
=
0
is a root then
k
=
2
If
x
=
2
is a root then
k
=
−
2
These two values of
k
split the real line into
3
parts:
(
−
∞
,
−
2
)
(
−
2
,
2
)
(
2
,
∞
)
Since we have observed that
f
(
x
)
has
3
zeros when
k
=
0
, we can deduce that:
(i) The given equation has
3
real roots when
k
∈
(
−
2
,
2
)
(ii) The given equation has
1
real root when
k
∈
(
−
∞
,
−
2
)
∪
(
2
,
∞
)