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Byju's Answer
Standard VIII
Mathematics
Factors of natural numbers
Without actua...
Question
Without actual division, prove that 2x
4
− 6x
3
+ 3x
2
+ 3x − 2 is exactly divisible by x
2
− 3x + 2.
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Solution
Let
f
x
=
2
x
4
-
6
x
3
+
3
x
2
+
3
x
-
2
and
g
x
=
x
2
-
3
x
+
2
.
g
x
=
x
2
-
3
x
+
2
=
x
2
-
2
x
-
x
+
2
=
x
x
-
2
-
1
x
-
2
=
x
-
1
x
-
2
Thus, f(x) is divisible by g(x) if f(1) = 0 and f(2) = 0. Now
f
1
=
2
1
4
-
6
1
3
+
3
1
2
+
3
1
-
2
=
2
-
6
+
3
+
3
-
2
=
0
f
2
=
2
2
4
-
6
2
3
+
3
2
2
+
3
2
-
2
=
32
-
48
+
12
+
6
-
2
=
0
Hence, 2x
4
− 6x
3
+ 3x
2
+ 3x − 2 is exactly divisible by x
2
− 3x + 2.
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