document.write( "Question 1116336: find a and b given that when x^3+2x^2+ax+6 is divided by x-1 the remainder is 4 and when divided by (x+2) the remainder is 16 \n" ); document.write( "

Algebra.Com's Answer #731252 by ikleyn(52779) $\"\"$ $\"About$

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\n" ); document.write( "This problem looks strange. It has only one unknown, but imposes two conditions on it.\r
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document.write( "The fact that  x^3+2x^2+ax+6  gives the remainder of 4 when divided by x-1  means that  the value of the polynomial at x= 1 is equal to 4\r\n" );
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document.write( "(according to the Remainder theorem):\r\n" );
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document.write( "1^3 + 2*1^2 + a*1 + 6 = 4,   or\r\n" );
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document.write( "1 + 2 + a + 6 = 4,   which implies\r\n" );
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document.write( "a = 4 - 1 - 2 - 6 = -5.\r\n" );
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document.write( "The fact that  x^3+2x^2+ax+6  gives the remainder of 16 when divided by x+2  means that  the value of the polynomial at x= -2 is equal to 16\r\n" );
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document.write( "(according to the Remainder theorem):\r\n" );
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document.write( "(-2)^3 + 2*(-2)^2 + a*(-2) + 6 = 16,   or\r\n" );
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document.write( "-8 + 8 - 2a + 6 = 16,   which implies\r\n" );
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document.write( "-2a = 16 + 8 - 8 - 6 = 10  ====>  a =  = -5.\r\n" );
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document.write( "Fortunately, both conditions give the same value for \"a\" equal to -5.\r\n" );
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document.write( "Answer.  a = -5.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Notice. The condition says \"find a and b\",   but the polynomial has only \"a\" and has no \"b\".\r
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\n" ); document.write( "\n" ); document.write( "On the Remainder theorem see the lessons \r
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\n" ); document.write( "\n" ); document.write( "    - Divisibility of polynomial f(x) by a binomial (x-a) and the Remainder theorem \r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on the Remainder theorem\r
\n" ); document.write( "\n" ); document.write( "    - OVERVIEW of lessons on Divisibility of polynomial f(x) by binomial (x-a) and the Remainder theorem\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
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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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