document.write( "Question 813301: Use the rational root theorem to find all rational roots of the function. Then find all actual roots.
\n" ); document.write( "3x^3+x^2-15x-5 \n" ); document.write( "

Algebra.Com's Answer #489579 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"3x%5E3%2Bx%5E2-15x-5\"$
\n" ); document.write( "The possible rational roots of a polynomial are all the ratios, positive and negative, that can be formed using a factor of the constant term in the numerator and a factor of the leading coefficient in the denominator.

\n" ); document.write( "Your constant term is 5. (Actually it is -5 but since we will be using all positive and negative ratios it doesn't make any difference if we use 5 or -5.) The factors of 5 are just 1 and 5.

\n" ); document.write( "Your leading coefficient is 3 whose factors are just 1 and 3.

\n" ); document.write( "Forming the ratios (as described above) we get the following list of possible rational roots:
\n" ); document.write( "+1/1, +5/1, +1/3 and +5/3.
\n" ); document.write( "The first two simplify:
\n" ); document.write( "+1, +5, +1/3 and +5/3.

\n" ); document.write( "Now we see which of these, if any, are actual roots. We'll use synthetic division to check. Checking 1:
\n" ); document.write( "

\r\n" );
document.write( "1  |   3   1   -15   -5\r\n" );
document.write( "----       3     4  -11\r\n" );
document.write( "      ------------------\r\n" );
document.write( "       3   4   -11  -16\r\n" );
document.write( "

The remainder is the -16 in the lower right corner. Since it is not zero 1 is not a root. Checking x = -1:
\n" ); document.write( "

\r\n" );
document.write( "-1 |   3   1   -15   -5\r\n" );
document.write( "----      -3     2   13\r\n" );
document.write( "      ------------------\r\n" );
document.write( "       3  -2   -13    8\r\n" );
document.write( "

The remainder is the 8. So -1 is not a root, either. But since this remainder is positive and the one we got for 1 was negative, there is a root somewhere in between. Since 8 is closer to zero than -16 I'm guessing that the root is closer to -1 than to 1. So I will try -1/3 next:
\n" ); document.write( "

\r\n" );
document.write( "-1/3 |   3   1   -15   -5\r\n" );
document.write( "------      -1     0    5\r\n" );
document.write( "        ------------------\r\n" );
document.write( "         3   0   -15    0\r\n" );
document.write( "

And we have a winner! -1/3 is a root (and 3x+1 is a factor). Not only that, but the rest of the bottom row tells us the other factor. The \"3 0 -15\" translates into $\"3x%5E2%2B0x-15\"$ or simply $\"3x%5E2-15\"$ . Since this is a quadratic we can find the remaining roots without having to try more rational roots. All we have to do is solve:
\n" ); document.write( " $\"3x%5E2-15=0\"$
\n" ); document.write( "Adding 15:
\n" ); document.write( " $\"3x%5E2=15\"$
\n" ); document.write( "Dividing by 3:
\n" ); document.write( " $\"x%5E2+=+5\"$
\n" ); document.write( "Square root of each side:
\n" ); document.write( " $\"x+=+0%2B-sqrt%285%29\"$
\n" ); document.write( "(Note: The zero is there only because algebra.com's formula drawing software will not let me use the \"plus or minus\" symbol without a number in front.)

\n" ); document.write( "So the three roots are -1/3 (rational), $\"sqrt%285%29\"$ (irrational) and $\"-sqrt%285%29\"$ (irrational) \n" ); document.write( "

\n" );