document.write( "Question 1036408: When f(x)is divided by x-1, the remainder is -1; when it is divided by $\"x%5E2\"$ , the remainder is -x-1. Find the remainder when f(x)is divided by $\"%28x%5E2%29%28x-1%29\"$ \n" ); document.write( "

Algebra.Com's Answer #651190 by robertb(5830) $\"\"$ $\"About$

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The remainder would be $\"x%5E2+-+x+-+1\"$ .\r
\n" ); document.write( "\n" ); document.write( "To see this, notice that the polynomial $\"f%28x%29+=+x%5E%282n-1%29+-+x+-+1\"$ , where $\"n%3E=2\"$ , will always give a remainder of -1 upon division by x-1 and a remainder of -x-1 upon division by $\"x%5E2\"$ . By using synthetic division with x - 1 as divisor, it can be easily seen that f(x) is unique in form. \r
\n" ); document.write( "\n" ); document.write( "By applying the usual polynomial division, the remainder after dividing $\"f%28x%29+=+x%5E%282n-1%29+-+x+-+1\"$ by $\"x%5E2%28x-1%29+=+x%5E3+-+x%5E2\"$ is $\"x%5E2+-+x+-+1\"$ . \n" ); document.write( "

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