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Given: $X_1$ and $X_2$ are the roots of the equation $ax^2+bx+c = 0$ $a\neq 0$

expand $X_1^3 + X_2^3$ using the parameters a,b and c

Here's what I tried to do:

$X_1^3 + X_2^3 = $ $(X_1\cdot X_2)\cdot(X_1^2-X_1\cdot X_2+X_2^2)$ From Viet's formulas: $(X_1\cdot X_2)(X_1^2-X_1\cdot X_2+X_2^2)$ = $\frac{-b}{a}\cdot(X_1^2+X_2^2- \frac{c}{a})=$ $\frac{c\cdot b}{a^2}\cdot(X_1^2+X_2^2)$

And this i all I can do the answer in my textbook is $\frac{3abc-b^3}{a^3}$

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HINT:

Use

$$p^3+q^3=(p+q)^3-3pq(p+q)$$


Alternatively,

$x^3=y$ and cubing $$(ax^2+bx)^3=(-c)^3$$

$$-c^3=a^3(x^3)^2+b^3(x^3)+3ab(x^3)(ax^2+bx)=a^3y^2+y(b^3-3abc)$$

$$a^3y^2+y(b^3-3abc)+c^3=0$$ whose roots are $x_1^3,x_2^3$

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  • $\begingroup$ Thank you, I solved it from the first hint. I will give you best answer after~7 minutes. $\endgroup$ Feb 8 '16 at 16:22

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