if α and β are the zeros of the polynomial f(x)=5x^2+4x−9 then evaluate the following: alpha^4-beta^4 About the author Camila
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 ) Reply
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
)