if alpha and beta are the zeros of the quadratic polynomial f(x)= ax²+bx+c, then evaluate alpha⁴-beta⁴
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Answer:
If α and β are zero of f(x)=ax
2
+bx+c. The evaluate α
4
×β
4
Note that we have
=(α+β)
4
=α
4
+β
4
+
4
C
1
α
3
β
1
+
4
C
2
α
2
β
2
+
4
C
3
α
1
β
1
3
Then, α
4
+β
4
=(α+β)
4
−4(α
3
β
1
+α
1
β
3
)−6α
2
β
2
=(α+β)
4
−4αβ(α
2
+β
2
)−6(αβ)
2
Maxover, we have
α+β=−
a
b
and αβ=
a
c
Then,
α
4
+β
4
=(−
a
b
)
4
−4(
a
c
)[(−
a
b
)
2
−2(
a
c
)]−6(
a
c
)
2
=
a
4
b
4
−
a
4c
(
a
2
b
2
)+8(
a
c
)
2
−6(
a
c
)
2
=
a
4
b
4
−
a
3
4b
2
c
+2
a
2
C
2
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