document.write( "Question 319996: rational zeroes theorem of x^4+2x^3-x^2+4x-6 \n" ); document.write( "
Algebra.Com's Answer #229172 by Edwin McCravy(20065)\"\" \"About 
You can put this solution on YOUR website!
\"x%5E4%2B2x%5E3-x%5E2%2B4x-6\"
\n" ); document.write( "
\r\n" );
document.write( "It's degree is 4, so it has four zeros. Now suppose the zeros were:\r\n" );
document.write( "A, B, C, and D.  Then it could be factored this way:\r\n" );
document.write( "\r\n" );
document.write( "\"%28x-A%29%28x-B%29%28x-C%29%28x-D%29\"\r\n" );
document.write( "\r\n" );
document.write( "then if somebody were to take a long time with a large sheet of paper and\r\n" );
document.write( "were to multiply that all the way, it would have sixteen terms before you\r\n" );
document.write( "started combining terms.  We are NOT going to do that, but only to ponder\r\n" );
document.write( "the question what If somebody actually took the trouble to multiply those\r\n" );
document.write( "four binomials all together.  If they did the last term would have to be\r\n" );
document.write( "ABCD.  Therefore the product of all the zeros has to be -6.  \r\n" );
document.write( " \r\n" );
document.write( "Since the leading coefficient is 1, it factors as above and therefore\r\n" );
document.write( "all feasible rational zeros are ± the factors of the absolute value of the\r\n" );
document.write( "totally numerical term, the last term, which is -6 and its absolute value is +6.\r\n" );
document.write( "The factors of 6 are these four integers: 1, 2, 3, and 6 itself. However\r\n" );
document.write( "it is possible that their opposites, or negatives, could be\r\n" );
document.write( "zeros also. So all the feasible zeros are these 8 possibilities:\r\n" );
document.write( "\r\n" );
document.write( "+1, +2, +3, +6, -1, -2, -3 and -6\r\n" );
document.write( "\r\n" );
document.write( "for which we usually just write:\r\n" );
document.write( "\r\n" );
document.write( "±1, ±2, ±3, ±6\r\n" );
document.write( "\r\n" );
document.write( "Let's find out if 1 is a zero.  That is the same thing as trying to find out if\r\n" );
document.write( "(x-1) is a factor of the polynomial.  So we divide it by (x-1), but instead of\r\n" );
document.write( "doing the long division like this:\r\n" );
document.write( "\r\n" );
document.write( "     _______________________\r\n" );
document.write( "x - 1)x4 + 2x3 - x2 + 4x - 6\r\n" );
document.write( "\r\n" );
document.write( "we do this synthetic division which is just a shortcut for getting the answer\r\n" );
document.write( "to that long division.\r\n" );
document.write( "\r\n" );
document.write( "    1|1  2  -1  4  -6\r\n" );
document.write( "     |   1   3  2   6 \r\n" );
document.write( "      1  3   2  6   0\r\n" );
document.write( "\r\n" );
document.write( "Which means you have factored \r\n" );
document.write( "\r\n" );
document.write( "\"x%5E4%2B2x%5E3-x%5E2%2B4x-6\" \r\n" );
document.write( "\r\n" );
document.write( "as\r\n" );
document.write( "\r\n" );
document.write( "\"%28x-1%29%28x%5E3%2B3x%5E2%2B2x%2B6%29\"\r\n" );
document.write( "\r\n" );
document.write( "Next we try to see if we can factor the 3rd degree\r\n" );
document.write( "polynomial \"x%5E3%2B3x%5E2%2B2x%2B6\".  It also ends with 6\r\n" );
document.write( "so it has those same feasible zeros.  So if 1 is a\r\n" );
document.write( "zero, then (x-1) will be a factor so we divide \"x%5E3%2B3x%5E2%2B2x%2B6\"  \r\n" );
document.write( "synthetically by x-1, and at the same time find out whether\r\n" );
document.write( "1 is a zero:\r\n" );
document.write( "\r\n" );
document.write( "1|1  3   2   6\r\n" );
document.write( " |   1   4   6 \r\n" );
document.write( "  1  4   6  12 \r\n" );
document.write( " \r\n" );
document.write( "No it isn't. So we have learned that 1 is not a zero and (x-1) is not \r\n" );
document.write( "a factor of the polynomial.\r\n" );
document.write( "\r\n" );
document.write( "So let's try to see if -1 is a zero, which is the same as seeing\r\n" );
document.write( "if (x+1) is a factor of \"x%5E3%2B3x%5E2%2B2x%2B6\".\r\n" );
document.write( "\r\n" );
document.write( "-1|1  3   2   6\r\n" );
document.write( "  |  -1  -2   0 \r\n" );
document.write( "   1  2   0   6\r\n" );
document.write( "\r\n" );
document.write( "No it isn't. So we have learned that -1 is not a zero and (x+1) is not \r\n" );
document.write( "a factor of the polynomial.\r\n" );
document.write( "\r\n" );
document.write( "So let's try to see if -3 is a zero, which is the same as seeing\r\n" );
document.write( "if (x+3) is a factor of \"x%5E3%2B3x%5E2%2B2x%2B6\".\r\n" );
document.write( "\r\n" );
document.write( "-3|1  3   2   6\r\n" );
document.write( "  |  -3   0  -6 \r\n" );
document.write( "   1  0   2   6\r\n" );
document.write( "\r\n" );
document.write( "Yes -3 is a zero. So we have learned that -3 is a zero and (x+3) is \r\n" );
document.write( "a factor of the polynomial.\r\n" );
document.write( "\r\n" );
document.write( "Which means we have so far factored \r\n" );
document.write( "\r\n" );
document.write( "\"x%5E4%2B2x%5E3-x%5E2%2B4x-6\" \r\n" );
document.write( "\r\n" );
document.write( "first as\r\n" );
document.write( "\r\n" );
document.write( "\"%28x-1%29%28x%5E3%2B3x%5E2%2B2x%2B6%29\"\r\n" );
document.write( "\r\n" );
document.write( "and now we have factored it further as\r\n" );
document.write( "\r\n" );
document.write( "\"%28x-1%29%28x%2B3%29%28x%5E2%2B0x%2B2%29\"\r\n" );
document.write( "\r\n" );
document.write( "or just\r\n" );
document.write( "\r\n" );
document.write( "\"%28x-1%29%28x%2B3%29%28x%5E2%2B2%29\"\r\n" );
document.write( "\r\n" );
document.write( "The last thing to factor is \"%28x%5E2%2B2%29\"\r\n" );
document.write( "\r\n" );
document.write( "When complex imaginary numbers are allowed we can factor the sum of\r\n" );
document.write( "two perfect squares, even though we could not do this when\r\n" );
document.write( "complex imaginary numbers were not allowed. To  factor that we realize\r\n" );
document.write( "that \"-i%5E2\" just equals 1. So we can multiply the 2 by \"-i%5E2\" without\r\n" );
document.write( "changing the value.  \r\n" );
document.write( "\r\n" );
document.write( "To factor \r\n" );
document.write( "\r\n" );
document.write( "\"%28x%5E2%2B2%29\"\r\n" );
document.write( "\r\n" );
document.write( "multiply the 2 by \"-i%5E2\"\r\n" );
document.write( "\r\n" );
document.write( "\"%28x%5E2%2B2%28-i%5E2%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"x%5E2-2i%5E2\"\r\n" );
document.write( "\r\n" );
document.write( "Now we know that \"%28sqrt%282%29%29%5E2=2\" so we can\r\n" );
document.write( "write the 2 as \"%28sqrt%282%29%29%5E2\", and we have:\r\n" );
document.write( "\r\n" );
document.write( "\"x%5E2-%28sqrt%282%29%29%5E2i%5E2\"\r\n" );
document.write( " \r\n" );
document.write( "Now it is the difference of two squares and we\r\n" );
document.write( "know how to factor it.\r\n" );
document.write( "\r\n" );
document.write( "\"%28x-sqrt%282%29%2Ai%29%28x%2Bsqrt%282%29%2Ai%29\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So the complete factorization of\r\n" );
document.write( "\r\n" );
document.write( "\"x%5E4%2B2x%5E3-x%5E2%2B4x-6\"\r\n" );
document.write( "\r\n" );
document.write( "is\r\n" );
document.write( "\r\n" );
document.write( "\"%28x-1%29%28x%2B3%29%28x-sqrt%282%29%2Ai%29%28x%2Bsqrt%282%29%2Ai%29\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "and the four zeros are 1, -3, \"sqrt%282%29%2Ai\" and \"-sqrt%282%29%2Ai\"\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );