document.write( "Question 1086582: When polynomial P(x) is divided by x + 1, x + 2, and x + 3, the remainders are 2, 3, and 6, respectively. Find the remainder when P(x) is divided by (x + 1)(x + 2)(x + 3). \n" ); document.write( "

Algebra.Com's Answer #700882 by ikleyn(52864) $\"\"$ $\"About$

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\n" ); document.write( "When polynomial P(x) is divided by x + 1, x + 2, and x + 3, the remainders are 2, 3, and 6, respectively.
\n" ); document.write( "Find the remainder when P(x) is divided by (x + 1)(x + 2)(x + 3).
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document.write( "We are given that \r\n" );
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document.write( "     \"when polynomial P(x) is divided by x + 1, x + 2, and x + 3, the remainders are 2, 3, and 6, respectively.\"\r\n" );
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document.write( "According to the Remainder theorem, it is equivalent to these equalities:\r\n" );
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document.write( "    P(-1) = 2,    (1)\r\n" );
document.write( "    P(-2) = 3,    (2)\r\n" );
document.write( "    P(-3) = 6.    (3)\r\n" );
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document.write( "Now, the question is to find a remainder polynomial R(x) after dividing P(x) by (x+1)*(x+2)*(x+3):\r\n" );
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document.write( "    P(x) = g(x)*(x+1)*(x+2)*(x+3) + R(x).     (4)\r\n" );
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document.write( "It is clear that the polynomial R(x) has the degree <= 2, so we can write\r\n" );
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document.write( "   R(x) = Ax^2 + Bx + C.                      (5)\r\n" );
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document.write( "Substituting x= -1, x= -2 and x= -3 into (4), from (1), (2) and (3) we have \r\n" );
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document.write( "    P(-1) = g(-1)*0 + R(-1) = 2,   i.e.   R(-1) = 2;     (6)\r\n" );
document.write( "    P(-2) = g(-2)*0 + R(-2) = 3,   i.e.   R(-2) = 3;     (7)\r\n" );
document.write( "    P(-3) = g(-3)*0 + R(-3) = 6,   i.e.   R(-3) = 6.     (8)\r\n" );
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document.write( "So, we need to find coefficients A, B and C of the remainder polynomial R(x) from conditions (6), (7) and (8).\r\n" );
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document.write( "Equation (6) gives\r\n" );
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document.write( "    A*(-1)^2 + B*(-1) + C = 2,   or   A - B + c = 2;     (9)\r\n" );
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document.write( "Equation (7) gives\r\n" );
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document.write( "    A*(-2)^2 + B*(-2) + C = 3,   or   4A - 2B + c = 3;   (10)\r\n" );
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document.write( "Equation (8) gives\r\n" );
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document.write( "    A*(-3)^2 + B*(-3) + C = 6,   or   9A - 3B + c = 6.   (11)\r\n" );
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document.write( "Thus you have this system of 3 equations to find A, B and C:\r\n" );
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document.write( " A -  B + c = 2,\r\n" );
document.write( "4A - 2B + c = 3,\r\n" );
document.write( "9A - 3B + c = 6.\r\n" );
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document.write( "Solve it by any method you want/you know (Substitution, Elimination, Determinanf (= Cramer's rule) ). You will get  A = 1,  B= 2  and  C = 3.\r\n" );
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document.write( "So, the remainder, which is under the question, is R(x) = x^2 + 2x + 3.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Answer. The remainder under the question is R(x) = x^2 + 2x + 3.\r
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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 lesson\r
\n" ); document.write( "\n" ); document.write( "    - Divisibility of polynomial f(x) by binomial x-a\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 lesson is 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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