SOLUTION: 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

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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
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
2x%5E3+%2B21x%5E2+%2Bmx+%2B+q+=0
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:
a%28x-r%29%28x-2r%29%28x-4r%29+=+0
We now multiply this out:
a%28x-r%29%28x%5E2-6xr%2B8r%5E2%29+=+0
a%28x%5E3-6x%5E2r%2B8xr-x%5E2r%2B6xr%5E2-8r%5E3%29+=+0
ax%5E3-6ax%5E2r%2B8axr%5E2-ax%5E2r%2B6axr%5E2-8ar%5E3%29+=+0
Adding like terms:
ax%5E3-7ax%5E2r%2B14axr%5E2-8ar%5E3%29+=+0
This is a general equation for a polynomial whose roots are r, 2r and 4r. Our equation:
2x%5E3+%2B21x%5E2+%2Bmx+%2B+q+=0
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:
%282%29x%5E3-7%282%29x%5E2r%2B14%282%29xr%5E2-8%282%29r%5E3%29+=+0
which simplifies as follows:
2x%5E3-14x%5E2r%2B28xr%5E2-16r%5E3%29+=+0

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:
2x%5E3-14x%5E2%28-3%2F2%29%2B28x%28-3%2F2%29%5E2-16%28-3%2F2%29%5E3%29+=+0
Simplifying...
2x%5E3-14x%5E2%28-3%2F2%29%2B28x%289%2F4%29-16%28-27%2F8%29+=+0
2x%5E3%2B21x%5E2%2B63x%2B54+=+0

We can now see that "m" must be 63 and "q" must be 54.

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.