document.write( "Question 1195206: If x+2 and x-3 are factors of the polynomial $\"p%28x%29=x%5E3%2B5x%5E2%2Bax%2Bb\"$ , find a. \n" ); document.write( "

Algebra.Com's Answer #827623 by ikleyn(52778) $\"\"$ $\"About$

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\n" ); document.write( "If x+2 and x-3 are factors of the polynomial $\"p%28x%29=x%5E3%2B5x%5E2%2Bax%2Bb\"$ , find a.
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document.write( "If (x+2) and (x-3) are factors of the polynomial  p(x)=x^3+5x^2+ax+b, it means,\r\n" );
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document.write( "due to the Remainder theorem, that x= -2 and x= 3 are the roots of the polynomial.\r\n" );
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document.write( "So, we substitute x= -2 and x= 3 into the polynomial and equate it to zero.\r\n" );
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document.write( "It gives us two equation for two unknowns \"a\" and \"b\"\r\n" );
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document.write( "    (-2)^3 + 5*(-2)^2 - 2a + b = 0    (1)\r\n" );
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document.write( "       3^3 + 5*3^2    + 3a + b = 0    (2)\r\n" );
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document.write( "Collect like terms and simplify (1) and (2)\r\n" );
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document.write( "     12 - 2a + b = 0    (3)\r\n" );
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document.write( "     72 + 3a + b = 0    (4)\r\n" );
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document.write( "Now subtract equation (4) from equation (3).  You will get\r\n" );
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document.write( "     60 + 5a     = 0,\r\n" );
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document.write( "which implies  5a = -60,  a = -60/5 = -12.\r\n" );
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document.write( "ANSWER.  a = -12.\r\n" );
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