Stitz-Zeager_College_Algebra_e-book

1 10 with a 09 10 we get an 10 10 for 9 n 0

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Unformatted text preview: ic in disguise, we write x4 + 16 = x2 2 + 4 2 = x2 2 + 8x2 + 42 − 8x2 = x2 + 4 2 − 8x2 528 Systems of Equations and Matrices and obtain a difference of two squares: x2 + 4 √ x4 + 16 = x2 + 4 − 2x 2 2 √2 and 8x2 = 2x 2 . Hence, √ √ x2 + 4 + 2x 2 = x2 − 2x 2 + 4 √ x2 + 2 x 2 + 4 The discrimant of both of these quadratics works out to be −8 < 0, which means they are irreducible. We leave it to the reader to verify that, despite having the same discriminant, these quadratics have different zeros. The partial fraction decomposition takes the form 8x2 Ax + B Cx + D 8x2 √ √ √ √ = + = 2 − 2x 2 + 4 x2 + 2 x 2 + 4 2 − 2x 2 + 4 2 + 2x 2 + 4 x4 + 16 x x x √ √ We get 8x2 = (Ax + B ) x2 + 2x 2 + 4 + (Cx + D) x2 − 2x 2 + 4 or √ √ √ √ 8x2 = (A + C )x3 + (2A 2 + B − 2C 2 + D)x2 + (4A + 2B 2 + 4C − 2D 2)x + 4B + 4D which gives the system √A + C √ 2A 2 √ B − 2C 2 + D + √ 4A + 2 B 2 + 4 C − 2D 2 4B + 4 D = = = = 0 8 0 0 We choose substitut...
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