if α and β are the zeros of the polynomial f(x)=5x^2+4x−9 then evaluate the following: alpha^4-beta^4
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Answer:
Answer
Correct option is
D
125
854
α and β are the zeros of the polynomial f(x)=5x
2
+4x−9
Here,
α+β=
a
−b
=
5
−4
αβ=
a
c
=
5
−9
Now,
(α+β)
2
=α
2
+β
2
+2αβ
⇒
5
−4
=α
2
+β
2
+2
5
−9
⇒
25
16
=α
2
+β
2
+
5
−18
⇒
25
16
+
5
18
=α
2
+β
2
⇒
25
16+90
=α
2
+β
2
⇒α
2
+β
2
=
25
106
Again,
(α−β)
2
=α
2
+β
2
−2αβ
⇒(α−β)
2
=
25
106
−2
5
−9
⇒(α−β)
2
=
25
106
+
5
18
⇒(α−β)
2
=
25
106+90
⇒(α−β)
2
=
25
196
⇒(α−β)=
5
14
Now,
(α
3
−β
3
)=(α−β)(α
2
+β
2
+αβ)
⇒(α
3
−β
3
)=(
5
14
)[
25
106
+(
5
−9
)]
⇒(α
3
−β
3
)=(
5
14
)(
25
106−45
)
⇒(α
3
−β
3
)=(
5
14
)(
25
61
)
⇒(α
3
−β
3
)=(
125
854
)
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