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Adv Alegbra HW Solutions 144

Adv Alegbra HW Solutions 144 - 144(ii Show that one root of...

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144 (ii) Show that one root of f ( X ) = X 3 + X 2 36 is an integer and fi nd the other two roots. Compare your method with Cardano s formula and with Vi`ete s trigonometric solution. Solution. Corollary 3.91 says that any integer root is a divisor of 36, and one checks that 3 is a root; therefore, X 3 + X 2 36 = ( X 3 )( X 2 + 4 X + 12 ). The quadratic formula gives the other two roots: 4 ± 8, both of which are negative. The other two methods are longer. Both require the substitution X = x 1 3 , yielding the reduced cubic x 3 1 3 x 970 27 . We do not give the calculations. 5.6 Show that if u is a root of a polynomial f ( x ) R [ x ] , then the complex conjugate u is also a root of f ( x ) . Solution. Let f ( x ) = r i x i . If 0 = r i u i , then since complex conju- gation is an isomorphism fi xing each real number, 0 = 0 = r i u i = r i u i = f ( u ). 5.7 Assume that 0 3 α < 360 . (i) If cos 3
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