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Question 20
If $$\alpha$$ and $$\beta$$ are the roots of equation $$x^2 - x + 1 = 0,$$ then which equation will have roots $$\alpha^3$$ and $$\beta^3?$$
Solution
$$x^2 - x + 1 $$= 0
$$\alpha\beta$$=1
$$\alpha+\beta$$=1
cubing on both sides
$$\alpha^3 + \beta^3+3\alpha\beta(\alpha+\beta)$$=1
$$\alpha^3 + \beta^3+3*1(1)$$=1
$$\alpha^3 + \beta^3$$=-2
$$\alpha^3\beta^3$$=1
Sum of the roots=-2
product=1
Required equation is $$x^2 + 2x + 1 = 0$$
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