20. Obtain all other zeroes of the polynomial x ^ 4 - 3sqrt(2) * x ^ 3 + 3x ^ 2 + 3sqrt(2) * x - 4 if two of its zeroes are sqrt(2) and 2sqrt(2)
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Answer:
given
⇒P(x)=x
4
−3
2
x
3
+3x
2
+3
2
x−4=0
as Two zero is given
2
,2
2
∴(x−
2
)(x−2
2
)=x
2
−3
2
x+4
Now considering a standard quadratic equation
ax
2
+bx+c=0
We can say
(x
2
−3
2
x+4)(ax
2
+bx+c)=p(x)
⇒ax
4
−3a
2
x
3
+4ax
2
+bx
3
−3b
2
x
2
+4bx+cx
2
−3c
2
x+4c=p(x)
⇒ax
4
−x
3
(3a
2
−b)+x
2
(4a−3b
2
+c)+x(4b−3c
2
)+4c=p(x)
Now comparing with P(x) we het
⇒4c=−4⇒4b−3c
2
=3
2
⇒3a
2
−b=3
2
⇒c=−1⇒4b=3
2
−3
2
⇒3a
2
=3
2
⇒b=0⇒a=1
∴ We can say other roots are
⇒x
2
−1=0
⇒x
2
=1
⇒x=±1=+1,−1 (Ans)
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