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Mth510Asgt2Solutions_W05

# Mth510Asgt2Solutions_W05 - Mathematics 510 Winter 2005...

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Mathematics 510 Winter 2005 Problem Set 2 Solutions Due: Thursday March 10, in class. 1. Plot graphs of the following polynomials and factor them completely into products of linear and quadratic factors. Use only methods taught in class to find the roots. Your plots should clearly show the real roots of the polynomial. Note: an acceptible plot is one that gives a good indication where the roots are. (a) P ( x ) = x 3 - 1 . 0 x 2 - 5 . 17 x + 5 . 566 = ( x + 2 . 3)( x - 1 . 1)( x - 2 . 2). (b) P ( x ) = x 3 - 5 . 9 x 2 + 9 . 35 x - 4 . 477 = ( x - 1 . 1) 2 ( x - 3 . 7). (c) P ( x ) = x 4 - 3 . 8 x 3 - 3 . 04 x 2 + 15 . 158 x - 9 . 4017 = ( x + 2 . 1)( x - 1 . 1) 2 ( x - 3 . 7). (d) P ( x ) = x 4 - 5 . 9 x 3 + 21 . 18 x 2 - 45 . 76 x + 33 . 80 = ( x - 1 . 3)( x - 2 . 6)( x 2 - 2 x + 10). (e) P ( x ) = x 5 - 6 . 13 x 4 + 22 . 537 x 3 - 50 . 6314 x 2 + 44 . 3248 x - 7 . 7740 = ( x - 2 . 3)( x - 1 . 3)( x - 2 . 6)( x 2 - 2 x + 10). 2. Find the eigenvalues and eigenvectors of the matrix: A = 1 2 - 4 2 - 2 - 2 - 4 - 2 1 . Do it once by hand and once with a software package, and compare your answers. The eigenvalues of A are 6 , - 3 , - 3 with corresponding eigenvectors being non-zero multiples of (2 , 1 , - 2), ( - 1 , 2 , 0) and (1 , 0 , 1) respectively.

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Mth510Asgt2Solutions_W05 - Mathematics 510 Winter 2005...

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