document.write( "Question 1183613: The expression ax^3 + bx^2 - 5x + 2a is exactly divisible by x^2 - 3x - 4. Calculate the value of a and of b and factorise the expression completely. \n" ); document.write( "

Algebra.Com's Answer #814028 by ikleyn(52944) $\"\"$ $\"About$

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\n" ); document.write( "The expression ax^3 + bx^2 - 5x + 2a is exactly divisible by x^2 - 3x - 4.
\n" ); document.write( "Calculate the value of a and of b and factorise the expression completely.
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\n" ); document.write( "\n" ); document.write( " Tutor @KMST provided very detailed long solution, covering different possible options.\r
\n" ); document.write( "\n" ); document.write( " I will try to give shorter solution in hope that it has its own charm.\r
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document.write( "The trinomial  x^2 - 3x - 4  is factorable:  x^2 - 3x - 4 = (x-4)*(x+1).\r\n" );
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document.write( "It means that the given polynomial of the degree 3,  ax^3 + bx^2 - 5x + 2a,  is divisible \r\n" );
document.write( "by both binomials  (x-4)  and  (x+1).\r\n" );
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document.write( "Due to the Remainder theorem, it means that the values x= 4  and  x= -1  are the roots of that polynomial.\r\n" );
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document.write( "So, we substitute the values x= 4  and  x= -1 into the given polynomial, equate it to zero and\r\n" );
document.write( "obtain two equations for the unknown coefficients  \"a\"  and  \"b\"\r\n" );
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document.write( "    a*4^3    + b*4^2 - 5*4    + 2a = 0      (1)\r\n" );
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document.write( "    a*(-1)^3 + b*1^2 - 5*(-1) + 2a = 0      (2)\r\n" );
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document.write( "Simplifying, you get\r\n" );
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document.write( "    64a + 16b - 20 + 2a = 0                 (1')\r\n" );
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document.write( "    -a  +   b +  5 + 2a = 0                 (2')\r\n" );
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document.write( "Simplifying further, you get\r\n" );
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document.write( "    66a + 16b = 20                          (1'')\r\n" );
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document.write( "      a +   b = -5                          (2'')\r\n" );
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document.write( "After solving the system, you get  a= 2,  b= -7.\r\n" );
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document.write( "Now the problem requires to find the third linear binomial, which is a third divisor to the given polynomial.\r\n" );
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document.write( "Use the Vieta's theorem:  the sum of the roots is equal to   =  = .\r\n" );
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document.write( "    so, we write  4 + (-1) + t = \r\n" );
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document.write( "where \"t\" is the third root, and we obtain from it\r\n" );
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document.write( "    t =  = .\r\n" );
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document.write( "Thus the third root is  , the associate binomial factor is  (x-1/2), and the required binomial decomposition is\r\n" );
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document.write( "    ax^3 + bx^2 - 5x + 2a = 2x^3 -7x^2 - 5x + 4 =  = (2x-1)*(x-4)*(x+1).\r\n" );
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