document.write( "Question 1159546: The function f is given by f(x)=x^3+2x^2+ax-8 where a is constant.when f(x) is divided by (x-2) the remainder is -6.
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Algebra.Com's Answer #782590 by ikleyn(52786) $\"\"$ $\"About$

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document.write( "According to the Remainder theorem, the fact that the remainder is -6, when f(x) is divided by (x-2), means that f(2) = -6.\r\n" );
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document.write( "In other words,  \r\n" );
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document.write( "    2^3 + 2*2^2 + a*2 - 8 = -6.\r\n" );
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document.write( "It implies\r\n" );
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document.write( "    2a = -6 - 2^3 - 2*2^2 + 8 = -14.\r\n" );
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document.write( "Hence,  a = -14/2 = -7.    \r\n" );
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document.write( "Thus the polynomial  f(x)  is\r\n" );
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document.write( "    f(x) = x^3 + 2x^2 - 7x - 8.    (1)\r\n" );
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document.write( "Now, let us check that f(-1) = 0.\r\n" );
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document.write( "For it, substitute x= -1 into the polynomial (1).  You will get\r\n" );
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document.write( "    f(-1) = (-1)^3 + 2*(-1)^2 - 7*(-1) - 8 = -1 + 2 + 7 - 8 = 0.\r\n" );
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document.write( "Now apply the Remainder theorem again.\r\n" );
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document.write( "It says that if  f(-1) = 0,  then f(x) is divisible by the binomial (x-(-1)) = (x+1).\r\n" );
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document.write( "It is EXACTLY what has to be proved.\r\n" );
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document.write( "The proof is completed.\r\n" );
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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
\n" ); document.write( "\"Divisibility of polynomial f(x) by binomial (x-a). The Remainder theorem\".\r
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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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