document.write( "Question 1209745: Fill in the blanks, to complete the factorization:\r
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\n" ); document.write( "\n" ); document.write( "(a^2 + b^2 - c^2)^2 - 4a^2 b^2 - 4a^2 c^2 + 4b^2 c^2 = (a + ___)(a + ___)(a + ___)(a + ___) \n" ); document.write( "

Algebra.Com's Answer #850112 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "(a^2 + b^2 - c^2)^2 - 4a^2 b^2 - 4a^2 c^2 + 4b^2 c^2 = (a + ___)(a + ___)(a + ___)(a + ___)
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document.write( "        Step 1.  Decompose into the product of two quadratic polynomials\r\n" );
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document.write( "    (a^2 + b^2 - c^2)^2 - 4a^2*b^2 - 4a^2*c^2 + 4b^2*c^2 =\r\n" );
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document.write( "  =  a^4 + b^4 + c^4 + 2a^2*b^2 - 2a^2*c^2 - 2b^2*c^2 - 4a^2*b^2 - 4a^2*c^2 + 4b^2*c^2 = \r\n" );
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document.write( "           next step make a routine combining like terms\r\n" );
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document.write( "  = a^4 + b^4 + c^4 - 2a^2*b^2 - 6a^2*c^2 + 2b^2*c^2 = \r\n" );
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document.write( "           next step make grouping/re-grouping\r\n" );
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document.write( "  = (a^4 + b^4 + c^4 2 - 2a^2*b^2 - 2a^2*c^2 + 2b^2*c^2) - 4a^2*c*2 = \r\n" );
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document.write( "           next step complete the squares\r\n" );
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document.write( "   = (-a^2 + b^2 + c^2)^2 - 4a^2*c^2 = \r\n" );
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document.write( "           next step factor as the difference of squares \r\n" );
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document.write( "   = (-a^2 + b^2 + c^2 - 2ac) * (-a^2 + b^2 + c^2 + 2ac) = \r\n" );
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document.write( "           next step is changing the signs everywhere in both parentheses \r\n" );
document.write( "           and light re-arranging in each parentheses (for further convenience)\r\n" );
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document.write( "   = (a^2 + 2ac - b^2 - c^2) * (a^2 - 2ac - b^2 - c^2).\r\n" );
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document.write( "        Step 2.  Decompose each parentheses as the product of linear binomials relative \"a\"\r\n" );
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document.write( "Now we want to decompose first parentheses (a^2 + 2ac - (b^2 + c^2)).    (1)\r\n" );
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document.write( "Consider this aggregate as a standard quadratic trinomial  a^2 + 2ac + X relative to variable 'a'.\r\n" );
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document.write( "Remember how to decompose a trinomial via its roots\r\n" );
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document.write( "     a^2 + 2ac + X = ,    (2)\r\n" );
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document.write( "where d is the discriminant.  In this case, the discriminant is\r\n" );
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document.write( "     d = (2c)^2 + 4*(b^2+c^2) = 4c^2 + 4b^2 + 4c^2 = 4(b^2+2c^2).\r\n" );
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document.write( "Therefore, decomposition for expression (1) takes the form\r\n" );
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document.write( "    a^2 + 2ac - (b^2+c^2) =  = \r\n" );
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document.write( "                          = .   (3)\r\n" );
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document.write( "Now, we want to decompose second parentheses (a^2 - 2ac - (b^2 + c^2)).    (4)\r\n" );
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document.write( "By analogy, consider this aggregate as a standard quadratic trinomial  a^2 - 2ac + X.\r\n" );
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document.write( "Remember how to decompose a trinomial via its roots\r\n" );
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document.write( "     a^2 - 2ac + X = ,    (5)\r\n" );
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document.write( "where d is the discriminant.  In this case, the discriminant is the same\r\n" );
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document.write( "     d = (2c)^2 + 4*(b^2+c^2) = 4c^2 + 4b^2 + 4c^2 = 4(b^2+2c^2).\r\n" );
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document.write( "Therefore, decomposition for expression (5) takes the form\r\n" );
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document.write( "    a^2 - 2ac - (b^2+c^2) = .    (6)\r\n" );
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document.write( "Combining everything above, we get finally this remarkable decomposition\r\n" );
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document.write( "    (a^2 + b^2 - c^2)^2 - 4a^2 b^2 - 4a^2 c^2 + 4b^2 c^2 = \r\n" );
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document.write( "  = .\r\n" );
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document.write( "which is the required form.\r\n" );
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document.write( "So, the four blanks are  ,  ,    and  .\r\n" );
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