College Algebra Exam Review 36

College Algebra Exam Review 36 - f;g;h 2 Kx , f C g D g C...

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46 1. ALGEBRAIC THEMES As an example of the sort of computations needed, let us verify the dis- tributive law: Let f.x/ D P ` i D 0 a i x i , g.x/ D P n j D 0 b j x j , and h.x/ D P n j D 0 c j x j . Then f.x/.g.x/ C h.x// D . ` X i D 0 a i x i /. n X j D 0 b j x j C n X j D 0 c j x j / D ` X i D 0 a i x i . n X j D 0 .b j C c j /x j / D ` X i D 0 n X j D 0 a i .b j C c j /x i C j D ` X i D 0 n X j D 0 .a i b j C a i c j /x i C j D ` X i D 0 n X j D 0 .a i b j /x i C j C ` X i D 0 n X j D 0 .a i c j /x i C j D . ` X i D 0 a i x i /. n X j D 0 b j x j / C . ` X i D 0 a i x i /. n X j D 0 c j x j / D f.x/g.x/ C f.x/h.x/: Verification of the remaining properties listed in the following propo- sition is left to the reader. Proposition 1.8.2. (a) Addition in KŒxŁ is commutative and associative; that is, for all
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Unformatted text preview: f;g;h 2 Kx , f C g D g C f; and f C .g C h/ D .f C g/ C h: (b) is an identity element for addition; that is, for all f 2 Kx , C f D f: (c) Every element f of Kx has an additive inverse f , satisfying f C . f / D 0:...
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