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

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