document.write( "Question 1170175: determine if p(x) is divisible by the given binomial
\n" ); document.write( "1.p(x)= x^5+32; x+2
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Algebra.Com's Answer #795030 by ikleyn(52812)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "In accordance with the Remainder theorem, it is enough to check if \r
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document.write( "(1)  the number -2 is the root of the first polynomial:\r\n" );
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document.write( "         indeed, \"%28-2%29%5E5+%2B+32\" = -32 + 32 = 0;\r\n" );
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document.write( "         so, -2 is the root; hence the polynomial is divisible by (x+2).\r\n" );
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document.write( "(2)  the number  \"sqrt%283%29\"  is the root of the second polynomial:\r\n" );
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document.write( "         indeed,  \"%28sqrt%283%29%29%5E3\" - \"3%2Asqrt%283%29\" = \"3%2Asqrt%283%29\" - \"3%2Asqrt%283%29\" = 0;\r\n" );
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document.write( "         so,  \"sqrt%283%29\"  is the root; hence the polynomial is divisible by \"x-sqrt%283%29\".\r\n" );
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\n" ); document.write( "\n" ); document.write( "The   ANSWER   is   \"YES\"   in both cases.\r
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\n" ); document.write( "\n" ); document.write( "   Theorem   (the remainder theorem)\r
\n" ); document.write( "\n" ); document.write( "   1. The remainder of division the polynomial  \"f%28x%29\"  by the binomial  \"x-a\"  is equal to the value  \"f%28a%29\"  of the polynomial. \r
\n" ); document.write( "\n" ); document.write( "   2. The binomial  \"x-a\"  divides the polynomial  \"f%28x%29\"  if and only if the value of  \"a\"  is the root of the polynomial  \"f%28x%29\",  i.e.  \"f%28a%29+=+0\".\r
\n" ); document.write( "\n" ); document.write( "   3. The binomial  \"x-a\"  factors the polynomial  \"f%28x%29\"  if and only if the value of  \"a\"  is the root of the polynomial  \"f%28x%29\",  i.e.  \"f%28a%29+=+0\".\r
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\n" ); document.write( "\n" ); document.write( "See the lessons\r
\n" ); document.write( "\n" ); document.write( "    - Divisibility of polynomial f(x) by binomial x-a and the Remainder theorem\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on the Remainder thoerem\r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic
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\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I
\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "to your archive and use it when it is needed.\r
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