document.write( "Question 708242: Write f(x) = x³ + 3x² + 36x + 108 as a product of linear factors. \n" ); document.write( "
Algebra.Com's Answer #436059 by Edwin Parker(36)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "f(x) = x³ + 3x² + 36x + 108\r\n" );
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document.write( "We can take x² out of the first two terms:\r\n" );
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document.write( "f(x) = x²(x + 3) + 36x + 108\r\n" );
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document.write( "We can take 36 out of the last two terms:\r\n" );
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document.write( "f(x) = x²(x + 3) + 36(x + 3)\r\n" );
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document.write( "We can take (x + 3) out of each of those two terms:\r\n" );
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document.write( "f(x) = (x + 3)(x² + 36)\r\n" );
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document.write( "We were told in an earlier algebra course that we\r\n" );
document.write( "can factor the difference of two squares, but we\r\n" );
document.write( "cannot factor the sum of two squares.  But that\r\n" );
document.write( "was before we had studied about imaginary numbers.\r\n" );
document.write( "But now we have learned that i² = -1 and so -i² = 1.\r\n" );
document.write( "So we can multiply the 36 by -i² without changing \r\n" );
document.write( "its value, so we get\r\n" );
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document.write( "f(x) = (x + 3)[x² + 36·(-i²)]\r\n" );
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document.write( "And now we have:\r\n" );
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document.write( "f(x) = (x + 3)(x² - 36i²)\r\n" );
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document.write( "And we do know how to factor the difference of two\r\n" );
document.write( "squares, so we get:\r\n" );
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document.write( "f(x) = (x + 3)(x - 6i)(x + 6i)\r\n" );
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document.write( "That's it.  The trick is to change the sum of two\r\n" );
document.write( "squares into the difference of two squares by multiplying\r\n" );
document.write( "the second square by -i², since that is just the same as \r\n" );
document.write( "multiplying by 1.\r\n" );
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document.write( "Edwin
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