document.write( "Question 71792: someone please show or explain the easiest way to understand this\r
\n" ); document.write( "\n" ); document.write( "Find the polynomial with leading coefficient 1 and degree 3 that has -1, 1, and 3 as roots.\r
\n" ); document.write( "\n" ); document.write( "Possible answers:\r
\n" ); document.write( "\n" ); document.write( "x3 - 3x2 -x +3
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\n" ); document.write( ".x3 +3x2 +x +3
\n" ); document.write( ".x3 + 3x2 +x +3\r
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Algebra.Com's Answer #51331 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
someone please show or explain the easiest way to understand this \r\n" );
document.write( "Find the polynomial with leading coefficient 1 and degree 3 that \r\n" );
document.write( "has -1, 1, and 3 as roots. \r\n" );
document.write( "Possible answers: \r\n" );
document.write( "x³ - 3x² - x + 3\r\n" );
document.write( "x³ - 3x² + x - 3\r\n" );
document.write( "x³ + 3x² + x + 3\r\n" );
document.write( "x³ + 3x² + x + 3 \r\n" );
document.write( "Thanks to anyone who can help me\r\n" );
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document.write( "Two methods.  The first method works only for multiple choice \r\n" );
document.write( "tests, but not on a test if you have to show your work.\r\n" );
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document.write( "The first way is to plug them all three, -1, 1, and 3, in and \r\n" );
document.write( "see which one gives you 0 for all three numbers \r\n" );
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document.write( "Plug x = -1 in the first answer:\r\n" );
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document.write( "   x³ - 3x² - x + 3\r\n" );
document.write( "(-1)³ - 3(-1)² - (-1) + 3\r\n" );
document.write( "    -1 - 3 + 1 + 3\r\n" );
document.write( "           0\r\n" );
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document.write( "Plug x = 1 in the first answer:\r\n" );
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document.write( "   x³ - 3x² - x + 3\r\n" );
document.write( "(1)³ - 3(1)² - (1) + 3\r\n" );
document.write( "   1 - 3 - 1 + 3\r\n" );
document.write( "         0\r\n" );
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document.write( "Plug x = 3 in the first answer:\r\n" );
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document.write( "   x³ - 3x² - x + 3\r\n" );
document.write( "(3)³ - 3(3)² - (3) + 3\r\n" );
document.write( "  27 - 27 - 1 + 3\r\n" );
document.write( "         0\r\n" );
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document.write( "Wow, they all three gave 0 for the answer, so that's the \r\n" );
document.write( "correct choice!\r\n" );
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document.write( "However, on another test it might not have been the first \r\n" );
document.write( "one.\r\n" );
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document.write( "If you plug -1, 1, and 3 in the 2nd choice you get \r\n" );
document.write( "-8,-4, and 0. They aren't all 0, so that's not the\r\n" );
document.write( "answer.\r\n" );
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document.write( "If you plug -1, 1, and 3 in the 3rd choice you get \r\n" );
document.write( "4, 8, and 60.  They aren't all 0, so that's not the \r\n" );
document.write( "answer.\r\n" );
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document.write( "Notice that the 4th choice is identical to the 3rd choice.\r\n" );
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document.write( "Now for method 2:\r\n" );
document.write( "This is the real way to work the problem. \r\n" );
document.write( "Solve the problem in reverse.\r\n" );
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document.write( "We begin with what we would have ended up with if we \r\n" );
document.write( "had solved it:\r\n" );
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document.write( "x = -1, x = 1, and x = 3\r\n" );
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document.write( "Now ask: What would we have set = 0 to get those, if we had\r\n" );
document.write( "solved it?\r\n" );
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document.write( "x + 1 = 0, x - 1 = 0, and x - 3 = 0\r\n" );
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document.write( "Next ask: Then what must the factored form of the polynomial \r\n" );
document.write( "equation have been to have given us that?\r\n" );
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document.write( "(x + 1)(x - 1)(x - 3) = 0\r\n" );
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document.write( "Next ask: What must the original polynomial equation have been?\r\n" );
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document.write( "Let's multiply the left side out and see\r\n" );
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document.write( "[(x + 1)(x - 1)](x - 3) = 0\r\n" );
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document.write( "[x² - x + x - 1)(x - 3) = 0\r\n" );
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document.write( "        (x² - 1)(x - 3) = 0\r\n" );
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document.write( "       x² - 3x² - x + 3 = 0\r\n" );
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document.write( "That's why it's the first choice.\r\n" );
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document.write( "Edwin
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