Construct an aquatic food chain with 5 trophic levels.
Answers
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Answer:
Use Vieta's method on 2x^2 + 2(m+n)x + m^2 + n^22x
2
+2(m+n)x+m
2
+n
2
.
\displaystyle\left \{ {{p + q=-(m + n)} \atop {pq =\dfrac{m^2 + n^2}{2}}} \right.
⎩
⎪
⎨
⎪
⎧
pq=
2
m
2
+n
2
p+q=−(m+n)
Two zeros of the new polynomial:
(p+q)^2=p^2+2pq+q^2(p+q)
2
=p
2
+2pq+q
2
and (p-q)^2=p^2-2pq+q^2(p−q)
2
=p
2
−2pq+q
2
Construct the new polynomial with Vieta's method.
Sum and product of the new polynomial:
Sum 2(p^2+q^2)2(p
2
+q
2
)
Product (p+q)^2(p-q)^2(p+q)
2
(p−q)
2
Finding the sum:
(p+q)^2-2pq=p^2+q^2(p+q)
2
−2pq=p
2
+q
2
=(m+n)^2-(m^2+n^2)=(m+n)
2
−(m
2
+n
2
)
=2mn=2mn
→ 4mn4mn is the sum.
Finding the product:
(p+q)^2-4pq=(p-q)^2(p+q)
2
−4pq=(p−q)
2
=(m+n)^2-2(m^2+n^2)=(m+n)
2
−2(m
2
+n
2
)
=-(m^2-2mn+n^2)=-(m-n)^2=−(m
2
−2mn+n
2
)=−(m−n)
2
→ -(m+n)^2(m-n)^2−(m+n)
2
(m−n)
2
is the product.
The new quadratic equation is x^2-4mnx-(m+n)^2(m-n)^2x
2
−4mnx−(m+n)
2
(m−n)
2
.
More information:
Vieta's Method
Consider a quadratic polynomial x^2+\dfrac{b}{a} x+\dfrac{c}{a}x
2
+
a
b
x+
a
c
.
If α and β are the zeroes of the polynomial then
(x-\alpha )(x-\beta )=x^2-(\alpha +\beta )x+\alpha \beta\;\textbf{[Factor Theorem]}(x−α)(x−β)=x
2
−(α+β)x+αβ[Factor Theorem] .
\alpha +\betaα+β is the sum of the two zeroes.
\alpha \betaαβ is the product of the two zeroes.
So \alpha +\beta =-\dfrac{b}{a}α+β=−
a
b
and \alpha \beta =\dfrac{c}{a}αβ=
a
c
.
this is answer for your upper question