document.write( "Question 1123636: Find all zeros using synthetic division. Show your work.\r
\n" ); document.write( "\n" ); document.write( "Q(x)= x^3 + (k+3)x^2 - k(2k-3)x - 6k^2 where k is a real number \n" ); document.write( "
Algebra.Com's Answer #740098 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "Q(x)= x³ + (k+3)x² - k(2k-3)x - 6k²\r\n" );
document.write( "\r\n" );
document.write( "Multiply out the coefficient -k(2k-3)\r\n" );
document.write( "\r\n" );
document.write( "Q(x)= x³ + (k+3)x² + (-2k²+3k)x - 6k²\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The potential zeros are ± the  factors of -6k², which are\r\n" );
document.write( "\r\n" );
document.write( "±1, ±2, ±3, ±6, ±k, ±2k, ±3k, ±6k, ±k², ±2k², ±3k², ±6k²\r\n" );
document.write( "\r\n" );
document.write( "That's a huge bunch to try! (lol), but I have a hunch that one\r\n" );
document.write( "of the zeros is k.  Let's see if I'm lucky:\r\n" );
document.write( "\r\n" );
document.write( " k | 1    k+3   -2k²+3k   -6k   \r\n" );
document.write( "   |      k      2k²+3k    6k    \r\n" );
document.write( "    ------------------------- \r\n" );
document.write( "     1   2k+3        6k     0\r\n" );
document.write( "\r\n" );
document.write( "The remainder is 0, so I was lucky! lol\r\n" );
document.write( "\r\n" );
document.write( "So k is a zero of Q(x), thus x-k is a factor\r\n" );
document.write( "of Q(x) and we have partially factored Q(x) as\r\n" );
document.write( "\r\n" );
document.write( "Q(x) = (x-k)[x² + (2k+3)x + 6k]\r\n" );
document.write( "\r\n" );
document.write( "So we set the quadratic factor equal zero and solve:\r\n" );
document.write( "\r\n" );
document.write( "x² + (2k+3)x + 6k = 0\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29+\"\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%28-%282k%2B3%29+%2B-+sqrt%28+%282k%2B3%29%5E2-4%281%29%286k%29+%29%29%2F%282%281%29%29+\"\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%28-2k-3+%2B-+sqrt%284k%5E2%2B12k%2B9-24k%29%29%2F2+\"\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%28-2k-3+%2B-+sqrt%284k%5E2-12k%2B9%29+%29%2F2+\"\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%28-2k-3+%2B-+sqrt%28%282k-3%29%5E2%29%29%2F2+\"\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%28-2k-3+%2B-+%282k-3%29%29%2F2+\"\r\n" );
document.write( "\r\n" );
document.write( "Using the + of ±\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%28-2k-3+%2B+%282k-3%29%29%2F2+\"\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%28-2k-3+%2B+2k-3%29%2F2+\"\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%28-6%29%2F2+\"\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+-3\", so -3 is a zero of Q(x)\r\n" );
document.write( "\r\n" );
document.write( "Using the - of ±\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%28-2k-3+-+%282k-3%29%29%2F2+\"\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%28-2k-3+-+2k%2B3%29%2F2+\"\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%28-4k%29%2F2+\"\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+-2k\", so -2k is a zero of Q(x)\r\n" );
document.write( "\r\n" );
document.write( "So the three zeros of Q(x) are k, -3 and -2k\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( " \n" ); document.write( "
\n" );