Answer to Question #245316 in Algebra for minu

Question

If m, n and p are the roots of the polynomial x³+5x-8 and sum of the roots is 0

0, then find the value of m³+n³+p^3.

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Answer 1

$m+n+p=-\dfrac{0}{1}=0$

$mn+np+mp=\dfrac{5}{1}=5$

$mnp=-\dfrac{8}{1}=-8$

$(m+n+p)^3=m^3+3m^2(n+p)+3m(n+p)^2+(n+p)^3$

$=m^3+3m^2n+3m^2p+3mn^2+6mnp+3mp^2$

$+n^3+3n^2p+3np^2+p^3$

$=(m^3+n^3+p^3)-3mnp+3m(mn+mp+np)$

$+3n(mn+mp+np)+3p(mn+mp+np)$

$=(m^3+n^3+p^3)-3(-8)+3m(5)+3n(5)$

$+3p(5)=(m^3+n^3+p^3)+24+15(m+n+p)$

$=(m^3+n^3+p^3)+24+15(0)=$

$=(m^3+n^3+p^3)+24=(0)^3$

Then

$m^3+n^3+p^3=-24$