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If $\alpha$ is a root of the equation $4x^2+2x-1=0$, then prove that $4\alpha^3-3\alpha$ is the other root. How do I proceed? The sum of the roots, the product of the roots lead me nowhere. Should I find the roots of the equation and substitute in the given two expressions of $\alpha$ and check whether manipulating the first root gives me the second root (which seems much complicated), or is there any other way?

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Check $4(4\alpha^3-3\alpha)^2+2(4\alpha^3-3\alpha)-1 = 0$ and $4\alpha^3-3\alpha \ne \alpha$. – njguliyev Nov 3 '13 at 12:03
up vote 2 down vote accepted

Set $x=4\alpha^2-3\alpha$ in the original polynomial equation, and check to see that the left side reduces to zero under the assumption that $4\alpha^2+2\alpha-1=0$, or equivalently that $\alpha^2 = \frac 14(-2\alpha+1)$.

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The other root $\beta$ is determined by the properties $\alpha+\beta=-\frac 12$ and $\alpha\beta=-\frac14$. Use polynomial division to show that $(4\alpha^3-3\alpha)+\alpha = [???]\cdot(4\alpha^2+2\alpha-1)-\frac12$ and $(4\alpha^3-3\alpha)\cdot\alpha = [???]\cdot(4\alpha^2+2\alpha-1)-\frac14$

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