If f(x) = 6x2 + 3 and g(x) = 4,
find (g – f )(x).
Answers
If f(x) = 6x2 + 3 and g(x) = 4,
find (g – f )(x).Put 2x+3=0, we get x=−
2
3
.
Substitute x=−
2
3
in f(x)=2x
3
+6x
2
+4x as follows:
f(x)=2x
3
+6x
2
+4x
f(−
2
3
)=2(−
2
3
)
3
+6(−
2
3
)
2
+4(−
2
3
)
f(−
2
3
)=2(−
8
27
)+6(
4
9
)−6
f(−
2
3
)=−
4
27
+
4
54
−6
f(−
2
3
)=
4
27
−6f(−
2
3
)=
4
27
−6
f(−
2
3
)=
4
3
Now substitute x=−
2
3
in g(x)=x
2
+3x+2 as follows:
g(x)=x
2
+3x+2
(−
2
3
)=(−
2
3
)
2
+3(−
2
3
)+2
g(−
2
3
)=
4
9
−
2
9
+2
g(−
2
3
)=
4
9−18+8
g(−
2
3
)=−
4
1
Finally substituting x=−
2
3
in the polynomial p(x)=f(x)+3g(x) we get,
p(x)=f(x)+3g(x)
p(−
2
3
)=f(−
2
3
)+3g(−
2
3
)
p(−
2
3
)=
4
3
+3(−
4
1
)p(−
2
3
)=0
Hence, the polynomial p(x)=f(x)+3g(x) is divisible by 2x+3.
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