4.4%20Polynomial%20Functions

4.4%20Polynomial%20Functions - Math 302 Modern Algebra...

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Math 302 Modern Algebra Spring 2009 Homework Solutions 4.4 POLYNOMIAL FUNCTIONS, ROOTS, AND IRREDUCIBILITY 3. Use the Remainder Theorem to determine if h ( x ) is a factor of f ( x ). (b) h ( x ) = x 1 2 and f ( x ) = 2 x 4 + x 3 + x 3 4 in Q [ x ] According to the Remainder Theorem, f ( a ) = 0 if and only if x a is a factor of f ( x ). Since f ( 1 2 ) = 2( 1 2 ) 4 + ( 1 2 ) 3 + 1 2 3 4 = 0 , then x 1 2 is a factor by the Remainder Theorem. (c) h ( x ) = x + 2 and f ( x ) = 3 x 5 + 4 x 4 + 2 x 3 x 2 + 2 x + 1 in Z 5 [ x ] Since f ( 2) = f (3) = 3(3 5 ) + 4(3 4 ) + 2(3 3 ) 3 2 + 2(3) + 1 = 1105 = 0 , then x + 2 is a factor by the Remainder Theorem. (d) h ( x ) = x 3 and f ( x ) = x 6 x 3 + x 5 in Z 7 [ x ] Since f (3) = 3 6 3 3 + 3 5 = 700 = 0 , then x 3 is a factor by the Remainder Theorem. 7. Use the Factor Theorem to show that x 7 x factors in Z 7 [ x ] as x ( x 1)( x 2)( x 3)( x 4)( x 5)( x 6), without doing any polynomial multiplication. Let f ( x ) = x 7 x . Then f (0) = 0 7 0 = 0 f (1) = 1 7 1 = 0 f (2) = 2 7 2 = 126 = 0 f (3) = 3 7 3 = 2184 = 0
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4.4%20Polynomial%20Functions - Math 302 Modern Algebra...

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