document.write( "Question 710429: The roots of 2x^3 +21X ^2 +mx + q =0 are in geometric progression with a ratio of 2. Find the values of m and q and the roots \n" ); document.write( "

Algebra.Com's Answer #437103 by jsmallt9(3758) $\"\"$ $\"About$

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$\"2x%5E3+%2B21x%5E2+%2Bmx+%2B+q+=0\"$
\n" ); document.write( "Let's say that a root is \"r\". Because of the geometric progression we are given, the other two roots could be expressed as 2r and 4r. We can write an equation for a polynomial with these roots:
\n" ); document.write( " $\"a%28x-r%29%28x-2r%29%28x-4r%29+=+0\"$
\n" ); document.write( "We now multiply this out:
\n" ); document.write( " $\"a%28x-r%29%28x%5E2-6xr%2B8r%5E2%29+=+0\"$
\n" ); document.write( " $\"a%28x%5E3-6x%5E2r%2B8xr-x%5E2r%2B6xr%5E2-8r%5E3%29+=+0\"$
\n" ); document.write( " $\"ax%5E3-6ax%5E2r%2B8axr%5E2-ax%5E2r%2B6axr%5E2-8ar%5E3%29+=+0\"$
\n" ); document.write( "Adding like terms:
\n" ); document.write( " $\"ax%5E3-7ax%5E2r%2B14axr%5E2-8ar%5E3%29+=+0\"$
\n" ); document.write( "This is a general equation for a polynomial whose roots are r, 2r and 4r. Our equation:
\n" ); document.write( " $\"2x%5E3+%2B21x%5E2+%2Bmx+%2B+q+=0\"$
\n" ); document.write( "must fit this pattern. Now we find the a, m and q that make our equation fit the pattern of the general equation. We have a term of 2x^3 and the general form has only one $\"x%5E3\"$ term, $\"ax%5E3\"$ . So a must be 2. Replacing the \"a\" in the general form with 2 we get:
\n" ); document.write( " $\"%282%29x%5E3-7%282%29x%5E2r%2B14%282%29xr%5E2-8%282%29r%5E3%29+=+0\"$
\n" ); document.write( "which simplifies as follows:
\n" ); document.write( " $\"2x%5E3-14x%5E2r%2B28xr%5E2-16r%5E3%29+=+0\"$

\n" ); document.write( "Now the modified general form has one $\"x%5E2\"$ term, $\"-14x%5E2r\"$ . This must match the $\"x%5E2\"$ term of our equation, $\"21x%5E2\"$ . Using this we can find that r must be -3/2 in order for $\"-14x%5E2r\"$ to be equal to $\"21x%5E2\"$ . Replacing the r in our modified general form with -3/2:
\n" ); document.write( " $\"2x%5E3-14x%5E2%28-3%2F2%29%2B28x%28-3%2F2%29%5E2-16%28-3%2F2%29%5E3%29+=+0\"$
\n" ); document.write( "Simplifying...
\n" ); document.write( " $\"2x%5E3-14x%5E2%28-3%2F2%29%2B28x%289%2F4%29-16%28-27%2F8%29+=+0\"$
\n" ); document.write( " $\"2x%5E3%2B21x%5E2%2B63x%2B54+=+0\"$

\n" ); document.write( "We can now see that \"m\" must be 63 and \"q\" must be 54.

\n" ); document.write( "The only thing left is the roots. We have already found that one root, r, is -3/2. The other two roots were 2r and 4r. Using -3/2 for we will find that the other roots are -3 and -6.
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