In this section we learn the complex conjugate root theorem for polynomials. This will allow us to find the zero(s) of a polynomial function in pairs, so long as the zeros are complex numbers.
Given a polynomial functions: \[f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_2x^2 + a_1x + a_0\] if it has a complex root (a zero that is a complex number), \(z\): \[f(z) = 0\] then its complex conjugate, \(z^*\), is also a root: \[f(z^*) = 0\]
The complex conjugate root theorem tells us that complex roots are always found in pairs. In other words if we find, or are given, one complex root, then we can state that its complex conjugate is also a root.
In the following tutorial we further explain the complex conjugate root theorem. We also work through an exercise, in which we use it. Indeed we look at the polynomial: \[f(x) = x^3 - 5x^2 + 17x- 13\] and are told \(2+3i\) is one of its roots. We then need to find all of its remaining roots and write this polynomial in its root-factored form.
Using the complex conjugate root theorem, find all of the remaining zeros (the roots) of each of the following polynomial functions and write each polynomial in root factored form:
Using the method shown in the tutorial above, answer each of the questions below.
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