| Reciprocal Roots |
Let K be a constant, and the following a polynomial equation in x:
x4 + Kx3 + 11x2 + 7x -12 = 0
What is the sum of the reciprocals of the roots of this equation? |
Problem Moderated by: Douglas |
| Problem Solution |
We want to find 1/a + 1/b + 1/c + 1/d, where a, b, c, and d are the roots of the equation. This may seem like an unlikely task, since we don't even know the value of K.
Let's begin by combining the fractions above into a single fraction:
(bcd + acd + abd + abc)/abcd.
The good news is, we can determine the product of the roots, without knowing what the roots are. It is the final coefficient (-12) divided by the first coefficient (1). So abcd = -12.
We can also determine the product of the roots taken three at a time: it is the opposite of the second to the last coefficient (-7), divided by the first coefficient (1).
So bcd + acd + abd + abc = -7.
Finally, the quotient of these two values is:
Note: the rule used to solve this problem works with any polynomial equation, but the signs are inverted for equations with an odd degree.
If you aren't sure why this rule works, try multiplying out the following, and you will quickly see how it works:
(x - a)(x - b)(x - c)(x - d) |
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