if alpha and beta are zeroes of polynomial x^2+3x-2,find the valu of 1/alpha^3+1/beta^3
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Step-by-step explanation:
ANSWER
Given that α & β are zero of polynomial
f(x)=x
2
−3x−2
therefore α+β=3
αβ=−2
Now, the zero of the required quadratic polynomial are,
2α+β
1
&
2β+α
1
Sum of the roots-
2α+β
1
+
2β+α
1
=
(2α+β)(2β+α)
2β+α+2α+β
=
4αβ+2α
2
+2β
2
+αβ
3(α+β)
=
4×(−2)+2[(α+β)
2
−2αβ]+(−2)
3×3
=
−10+2[9+2×2]
9
=
−10+26
9
=
16
9
Products of roots:-
2α+β
1
×
2β+α
1
=
4αβ+2[(α+β)
2
−2αβ]+αβ
1
=
16
1
Now Req eq.
x
2
−(sum of roots)x+ Product of roots=0
=x
2
−
16
9
x+
16
1
=0
=16x
2
−9x+16=
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