X3-2x2-5x+6 factorise
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Answered by
4
Answer:
Let
f
(
x
)
=
x
3
−
2
x
2
−
5
x
+
6
First note that the sum of the coefficients is
0
, so
x
=
1
is a zero of
f
(
x
)
...
f
(
1
)
=
1
−
2
−
5
+
6
=
0
Divide
f
(
x
)
by
(
x
−
1
)
to find:
f
(
x
)
=
x
3
−
2
x
2
−
5
x
+
6
=
(
x
−
1
)
(
x
2
−
x
−
6
)
Then
x
2
−
x
−
6
=
(
x
−
3
)
(
x
+
2
)
So the other two roots of
f
(
x
)
=
0
are
x
=
3
and
x
=
−
2
Alternatively, use the rational roots theorem to know that any rational roots of
f
(
x
)
=
0
must be of the form
p
q
where
p
,
q
are integers,
q
≠
0
,
p
a factor of the constant term
6
and
q
a factor of the coefficient
1
of the term (
x
3
) of highest degree. So the only possible rational roots are:
±
1
,
±
2
,
±
3
and
±
6
Try each to find
f
(
1
)
=
0
,
f
(
−
2
)
=
0
and
f
(
3
)
=
0
.
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