pp2a_5 - Math 293 Practice Prelim 2 Fall 2007 5 Solution At...

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Math 293 Practice Prelim 2 Fall 2007 5. Solution : At first, we need write down the characteristic equation: r 5 - r 3 + 2 r 2 + 2 r = 0 . (1) This is a polynomial of degree 5 , we don’t have a formula to get its roots immediately, we need factor it: r 5 - r 3 + 2 r 2 + 2 r = r ( r 4 - r 2 + 2 r + 2) = r [ r 2 ( r 2 - 1) + 2( r + 1)] = r [ r 2 ( r - 1)( r + 1) + 2( r + 1)] = r ( r + 1)[ r 2 ( r - 1) + 2] . The polynomial r 2 ( r - 1) + 2 = r 3 - r 2 + 2 is of degree 3 , there is a formula to get its roots, but we don’t use it here. Notice that r = - 1 is a root of r 3 - r 2 + 2 = 0 , so we can divide the polynomial
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