Find all the zeros of 2x4-9x3+5x2+3x-1 if two its zeros are 2+ root 3, 2- root 3
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Answer:
It is given that 2+
3
,2−
3
are two zeroes of the polynomial f(x)=2x
4
−9x
3
+5x
2
+3x−1
∴(x−(2+
3
))(x−(2−
3
))=(x−2−
3
)(x−2+
3
)
=(x−2)
2
−(
3
)
2
=x
2
−4x+4−3
=x
2
−4x+1
∴(x−(2+
3
))(x−(2−
3
))=x
2
−4x+1 is a factor of f(x)
Now divide f(x) by x
2
−4x+1
∴f(x)=(x
2
−4x+1)(2x
2
−x−1)
Hence, the other two zeroes of f(x) are the zeroes of the polynomial 2x
2
−x−1
2x
2
−x−1=(2x+1)(x−1)
Hence, the other two zeroes are −
2
1
,1.
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