Finding Polynomial Equation

[Caldus]

Sorry to keep asking questions...my teacher is horrible...

Anyway, I need to find a polynomial equation with degree 8 and contains the roots (3 - 2i), (-5 + 7i), 0 (with multiplicity 3), -3, and 4 (with multiplicity 2).

This is what I came up with:

p(x) = (x - (3 - 2i))(x - (-5 + 7i))(x)^3(x + 3)(x - 4)^2

Two things I'm wondering about (besides whether I got it right of course):

1. Don't I need to add in the conjugates for 3 - 2i and -5 + 7i in the equation? In that case, the degree would have to be 10, no?

2. Can I simplify this any further?

Thanks again.

 

[Hurkyl]

You only need to include the complex conjugates if you are looking for a real polynomial; they're not necessary for complex polynomials.

 

[tacman]

No need to apologize for asking questions, it's always better to clarify and make sure you understand something than to remain confused. As for your polynomial equation, you are correct in using the conjugate roots for 3 - 2i and -5 + 7i. This will ensure that the coefficients of the polynomial are all real numbers. However, adding the conjugates will not change the degree of the polynomial. The degree will still be 8 since the roots are still the same, it's just that the factors will be slightly different.

To simplify the polynomial further, you can use the FOIL method to expand the factors and combine like terms. This will give you a polynomial in standard form, where the terms are arranged in descending order of degree. However, it is not necessary to simplify the polynomial any further as it is already in its most simplified form.

In the future, if you are unsure about your answer, you can always use a graphing calculator to graph the polynomial and see if the given roots are indeed the x-intercepts. This will help you check your work and ensure that you have the correct polynomial equation. Keep up the good work and don't hesitate to ask for help when needed. Good luck!