Level of Question: Secondary (12)
Asking Question: Student
Write an equation for a polynomial of degree 5 given the following zeros:
-7+3i, -2+sqrt(13), 6
Thank you for your help, time, and effort. THANKS!!
If a polynomial in x has a root at x = a then [x - a] is a factor of the polynomial. You have roots at -7+3i, -2+sqrt(13) and 6, thus
are factors of the polynomial. Thus,
[x + 7 - 3i][x + 2 - sqrt(13)][x - 6]
is a polynomial with the three roots you want, but the polynomial is of degree 3 rather than 5.
- [x - (-7 + 3i)] = [x + 7 - 3i]
- [x - (-2 + sqrt(13))] = [x + 2 - sqrt(13)] and
- [x - 6]
You can write a polynomial with degree 5 that has these roots in many ways, for example
x2 [x + 7 - 3i][x + 2 - sqrt(13)][x - 6].
However I don't think that this is the answer that is expected in your textbook or by your teacher. If you expand the exprression above you will find that the coefficients are not integers, in fact they are not all real numbers. If the coefficients of a polynomial are integers then the roots come in conjugate pairs. In your example this means that, since -7 + 3i is a root so is -7 - 3i and since -2 + sqrt(13) is a root so is -2 - sqrt(13). Hence a degree 5 polynomial with integer coefficients and the roots you want is
[x + 7 - 3i][x + 7 + 3i][x + 2 - sqrt(13)][x + 2 + sqrt(13)][x - 6]
You should expand this expression to verify that the coefficients are integers.