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Let $a,$ $b,$ $c,$ $d,$ and $e$ be the distinct roots of the equation $x^5 + 7x^4 - 2 = 0.$ Find \begin{align*} &\frac{a^4}{(a - b)(a - c)(a - d)(a - e)} + \frac{b^4}{(b - a)(b - c)(b - d)(b - e)} \\ &\quad + \frac{c^4}{(c - a)(c - b)(c - d)(c - e)} + \frac{d^4}{(d - a)(d - b)(d - c)(d - e)} \\ &\quad + \frac{e^4}{(e - a)(e - b)(e - c)(e - d)}. \end{align*}

 
 Jan 8, 2021
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The roots of x^5 + 7x^4 - 2 = 0 are

r[1] = -6.99917

r[2] = -0.752186

r[3] = 0.0188858 - 0.729393i

r[4] = 0.0188858 + 0.729393i

r[5] = 0.713581

 

Plugging these value into the expression, we get an answer of 17.

 
 Jan 8, 2021

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