document.write( "Question 1049424: Find constants a and b, so x^3+ax+b is divisible y x^2+2x-2 \n" ); document.write( "

Algebra.Com's Answer #665044 by ikleyn(52878) $\"\"$ $\"About$

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\n" ); document.write( "Find constants a and b, so x^3+ax+b is divisible y x^2+2x-2
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document.write( "If  x^3+ax+b is divisible by x^2+2x-2, then \r\n" );
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document.write( "    x^3+ax+b = (x^2+2x-2)*(x+c),\r\n" );
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document.write( "where \"c\" is an unknown number. Now open the parentheses on the right. You will get\r\n" );
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document.write( "    x^3+ax+b = x^3 + 2x^2 - 2x + cx^2 + 2cx - 2c,  or\r\n" );
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document.write( "    x^3+ax+b = x^3 + (2+c)x^2 + (-2+2c)x - 2c.\r\n" );
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document.write( "Next, comparing the coefficients at x^2, x and the constant terms on both sides, you have these equalities\r\n" );
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document.write( "0 = 2 + c,       (1)       (coefficients at x^2)\r\n" );
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document.write( "a = -2 + 2c,     (2)       (coefficients at x)\r\n" );
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document.write( "b = -2c.         (3)       (constant terms).\r\n" );
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document.write( "Now from (1) you have c = -2;  from (2) you have a = -2+2*(-2) = -6;  and from (3) you have b = -2*(-2) = 4.\r\n" );
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document.write( "The summary is:  in order for x^3+ax+b be divisible by x^2+2x-2,\r\n" );
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document.write( "                 the following requirements must be in place: a = -6 and b = 4.  (They are necessary and sufficient conditions). \r\n" );
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document.write( "                 Then  c = -2, and  indeed x^3 -6x + 4 is divisible by x^2+2x-2.\r\n" );
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