document.write( "Question 242621: How do I find all rational roots of equation;
\n" ); document.write( "x^4+16x^3+96x^2+256x+256 \n" ); document.write( "
Algebra.Com's Answer #177609 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
\"x%5E4%2B16x%5E3%2B96x%5E2%2B256x%2B256\"
\n" ); document.write( "is not an equation. I assume it is supposed to be:
\n" ); document.write( "\"x%5E4%2B16x%5E3%2B96x%5E2%2B256x%2B256+=+0\"

\n" ); document.write( "To find the roots we will need to factor the left side. The expression on the left side:
  • Does not have a GCF (other than 1).
  • Does not fit any of the factoring patterns.
  • Is not a trinomial
  • Has an odd number of terms so it does not look like factoring by grouping will be possible.

\n" ); document.write( "So it looks like the only way to factor this is by trial and error of the possible rational roots.

\n" ); document.write( "There may be a lot of factors of 256. But two things will help shorten the process:
  • Since all the coefficients are positive, we cannot have any positive roots. After all, raising a positive number to a power and then multiplying it by a positive you will always get a positive. And if we add nothing but positive numbers, how can we get zero. So we know that we only need to try negative numbers. This cuts the number of possible roots in half.
  • 256 is a power of 2. So all its possible factors are also powers of 2.

\n" ); document.write( "So we are going to see if any negative powers of 2 are rational roots. We will do this using Synthetic Division:
\n" ); document.write( "
\r\n" );
document.write( "-2 |  1   16   96   256   256\r\n" );
document.write( "----      -2  -28  -136  -240\r\n" );
document.write( "     ------------------------\r\n" );
document.write( "      1   14   68   120   -16\r\n" );
document.write( "Nope.\r\n" );
document.write( "\r\n" );
document.write( "-4 |  1   16   96   256   256\r\n" );
document.write( "----      -4  -48  -192  -256\r\n" );
document.write( "     ------------------------\r\n" );
document.write( "      1   12   48    64     0\r\n" );
document.write( "

\n" ); document.write( "Yes. So now we have:
\n" ); document.write( "\"%28x%2B4%29%28x%5E3+%2B+12x+%2B+48+%2B+64%29+=+0\"
\n" ); document.write( "We need to keep looking. -2 did not work earlier. I cannot work now either. But -4 might work again (on (x^3 + 12x + 48 + 64)}}}:
\n" ); document.write( "
\r\n" );
document.write( "-4 |  1   12   48    64\r\n" );
document.write( "----      -4  -32   -64\r\n" );
document.write( "     ------------------------\r\n" );
document.write( "      1    8   16     0\r\n" );
document.write( "

\n" ); document.write( "Yes, again! So no we have:
\n" ); document.write( "\"%28x%2B4%29%28x%2B4%29%28x%5E2+%2B+8x+%2B+16%29+=+0\"
\n" ); document.write( "Since \"%28x%5E2+%2B+8x+%2B+16%29\" is a trinomial we can use either trinomial factoring or patterns to factor this into:
\n" ); document.write( "\"%28x%2B4%29%28x%2B4%29%28x%2B4%29%28x%2B4%29+=+0\"
\n" ); document.write( "or
\n" ); document.write( "\"%28x%2B4%29%5E4+=+0\"
\n" ); document.write( "So there is one root (of multiplicity 4): -4. \n" ); document.write( "
\n" );