document.write( "Question 1111313: the remainders when f(x)=x³+ax²+bx+c is divided by (x-1),(x+2) and (x-2) are respectively 2,-1 and 15, find the quotient and remainder when f(x) is divided by (x+1). \n" ); document.write( "

Algebra.Com's Answer #726295 by ikleyn(52790) $\"\"$ $\"About$

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document.write( "I will use the Remainder theorem. I will make all necessary explanations and references, but will not go in details.\r\n" );
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document.write( "Based on the Remainder theorem, from the given part you have these equations\r\n" );
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document.write( "f(1)  =  2,   or  1^3    + a*1^2    + b*1    + c =  2    (1)\r\n" );
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document.write( "f(-2) = -1,   or  (-2)^3 + a*(-2)^2 + b*(-2) + c = -1    (2)\r\n" );
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document.write( "f(2)  = 15,   or  2^3    + a*2^2    + b*2    + c = 15    (3)\r\n" );
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document.write( "Simplifying\r\n" );
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document.write( " 1 +  a +  b + c =  2       (1')\r\n" );
document.write( "-8 + 4a - 2b + c = -1       (2')\r\n" );
document.write( " 8 + 4a + 2b + c = 15       (3')\r\n" );
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document.write( "Simplifying one more time\r\n" );
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document.write( "      a +  b + c = 1        (1'')\r\n" );
document.write( "     4a - 2b + c = 7        (2'')\r\n" );
document.write( "     4a + 2b + c = 7        (3'')\r\n" );
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document.write( "Subtract (2'') from (3''). You will get  4b = 0  ====>  b = 0.\r\n" );
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document.write( "Now substitute this value of b into eqs (1'')  and (2'').  You will get\r\n" );
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document.write( "     a + c = 1        (4)\r\n" );
document.write( "    4a + c = 7        (5)\r\n" );
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document.write( "--------------------------------------- Subtract (4) from (5)\r\n" );
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document.write( "          3a = 6  ====>  a = 2\r\n" );
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document.write( "Then  from (4)  c = 1 - 2 = -1\r\n" );
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document.write( "Thus I restored the 3-rd degree polynomial. It is\r\n" );
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document.write( "f(x) = .\r\n" );
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document.write( "The rest is pure mechanical work:\r\n" );
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document.write( "f(x) = (x+1)*(x^2 + x -1).\r\n" );
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document.write( "Answer.  The quotient under the question is  (x^2 + x - 1).  The remainder is  0.\r\n" );
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\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
\n" ); document.write( "\n" ); document.write( "in this site.\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
\n" ); document.write( "
\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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