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Let \(r_1, \) \(r_2,\) \(\dots\) \(r_7\) be the roots of \(f(x) = x^7 + 5x^5 - 4x^4 + 5x^3 - 9x^2 + x + a,\)where \(a\) is some real constant. Let \(g(x) = x^3 + 3x - 3.\) Then the product \(g(r_1) g(r_2) \dotsm g(r_7)\) can be expressed as a polynomial in \(a.\) Find this polynomial.

 

Any help would be appreciated!!!

 Dec 27, 2021
 #1
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The polynomial is 3a^3 - 8a^2 + 11a - 17.

 Jan 9, 2022

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