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Given that x=2 is a root of p(x)=x^4-3x^3-18x^2+90x-100, find the other roots (real and nonreal).

 
 Jan 9, 2022
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By synthetic division, we get that \(p(x) = (x - 2)(x^3 - x^2 - 20x + 50)\). By Rational Root Theorem, the possible roots of \(x^3 - x^2 - 20x + 50\) are \(\pm 1, \pm 2, \pm 5, \pm 10, \pm 25\), and \(\pm 50\). Testing values, we get that \(x = -5\) works, so by synthetic division, we get that \(p(x) = (x - 2)(x + 5)(x^2 - 6x + 10)\).

 

By the quadratic formula, the roots of \(x^2 - 6x + 10\) are \(\frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 10}}{2 \cdot 1} = \frac{6 \pm 2i}{2} = 3 \pm i\). Thus, the roots of \(p(x)\) are \(\boxed{-5, 2, 3 + i, \text{and } 3 - i}\).

 
 Jan 10, 2022

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