document.write( "Question 1148165: For the function below neatly solve for x.
\n" ); document.write( "𝑓(𝑥) = 6𝑥^3 + 25𝑥^2 − 24𝑥 + 5
\n" ); document.write( "Use the Remainder Theorem to check potential rational zeros. (Remember the
\n" ); document.write( "techniques from the notes; you do not need to check ALL potential zeros.)
\n" ); document.write( "Use Synthetic Division to find a depressed polynomial once you know a zero. Repeat with additional zeros until you have the function down to a factorable function. \n" ); document.write( "
Algebra.Com's Answer #769527 by greenestamps(13200)\"\" \"About 
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\n" ); document.write( "The possible rational roots are of the form plus-or-minus p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

\n" ); document.write( "You can make the list as easily as we can. p is either 1 or 5; q is either 1, 2, 3, or 6.

\n" ); document.write( "You can go through the list of possible rational roots and try each one using synthetic division; but that is tedious and not a good use of your time.

\n" ); document.write( "One way to start on the problem and make much better use of your time (still giving you practice with synthetic division) is to use a graphing calculator to find one of the roots and go from there.

\n" ); document.write( "Or you can do some logical reasoning, using Vieta's Theorem to see that the sum of the roots is -25/6 and the product of the roots is -5/6.

\n" ); document.write( "That suggests that -5 is one of the roots; and synthetic division confirms that:
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document.write( "    -5  |  6  25  -24   5\r\n" );
document.write( "        |    -30   25  -5\r\n" );
document.write( "        +-----------------\r\n" );
document.write( "           6  -5    1   0

\n" ); document.write( "So

\n" ); document.write( "\"6x%5E3%2B25x%5E2-24x%2B5+=+%28x%2B5%29%286x%5E2-5x%2B1%29\"

\n" ); document.write( "Then the quadratic factors nicely:\n" ); document.write( "\"6x%5E3%2B25x%5E2-24x%2B5+=+%28x%2B5%29%283x-1%29%282x-1%29\"

\n" ); document.write( "And so the roots are -5, 1/3, and 1/2.
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