find the square root by division method 4+25x^2-12x-24x^3+16x^4
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Step-by-step explanation:
Explanation:
Given:
4
+
25
x
2
−
12
x
−
24
x
3
+
16
x
4
Rearrange in standard form with descending powers of
x
:
16
x
4
−
24
x
3
+
25
x
2
−
12
x
+
4
Note that if this polynomial has a square factor then that will also be a factor of its derivative:
64
x
3
−
72
x
2
+
50
x
−
12
=
2
(
32
x
3
−
36
x
2
+
25
x
−
6
)
So we want to find the GCF of these polynomials.
Multiplying the original quartic by
2
, we get:
32
x
4
−
48
x
3
+
50
x
2
−
24
x
+
8
=
x
(
32
x
3
−
36
x
2
+
25
x
−
6
)
−
12
x
3
+
25
x
2
−
18
x
+
8
Now:
3
(
32
x
3
−
36
x
2
+
25
x
−
6
)
+
8
(
−
12
x
3
+
25
x
2
−
18
x
+
8
)
=
96
x
3
−
108
x
2
+
75
x
−
18
−
96
x
3
+
200
x
2
−
144
x
+
64
=
92
x
2
−
69
x
+
46
=
23
(
4
x
2
−
3
x
+
2
)
Then we find:
(
4
x
2
−
3
x
+
2
)
2
=
16
x
4
−
24
x
3
+
25
x
2
−
12
x
+
4
as required.
So:
√
16
x
4
−
24
x
3
+
25
x
2
−
12
x
+
4
=
4
x
2
−
3
x
+
2
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