Math CentralQuandaries & Queries


Question from aydee, a student:

1. find the equation for the roots given: 7,-5,2

2. find the equation for the roots given: -5,2+sqrt3,2-sqrt3

Aydee, we have two responses for you:

Hi Aydee,

If you start with the equation x3 - 4x2 - 7x + 10 = 0 you can factor it and get (x - 1)(x + 2)(x - 5) = 0 and thus the roots are x = 1, x = -2 and x = 5. To solve your problems you apply this process backwards.

Find a polynomial with roots 1, -2 and 5. Start with the roots x = 1, x = -2 and x = 5 and construct the polynomial (x - 1)(x + 2)(x - 5) = 0. You can then expand this expression if you wish and get x3 - 4x2 - 7x + 10 = 0.




The roots of an equation are the values that make it equal zero. If this is a regular polynomial, then that means there are as many factors (at least) as there are roots. So the equation is the product of three factors if there are three roots. Each root corresponds to one of the factors equalling zero, so you can deal with them individually. Think of each of the roots as a separate function if you like:

f(x)g(x)h(x) = 0, so f(x) = 0 or g(x) = 0 or h(x) = 0

Since it doesn't matter which is which, let's say f(7) = 0 (7 is one of the roots of your first equation). What is the simplest value thing you can do to 7 to turn it into 0? Subtract 7. So f(x) = x - 7. See how if x = 7, then f(x) = 0? And this means f(7)g(7)h(7) must be 0 as well. Hence, 7 is a root and one of the factors of the equation is (x-7).

Hope this helps,
Stephen La Rocque.


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