It looks like you have an equation involving unknowns $a$, $b$, $c$, and $d$, and you are provided with some roots of the equation: $1 + 3i$ and $\frac{1}{2}$. This equation is likely a cubic or quartic polynomial with these roots. To proceed, we will use these roots to help determine the values of $a$, $b$, $c$, and $d$.

The general form of the equation is:

$a x^3 + b x^2 + c x + d = 0$

If we assume that the roots are $1 + 3i$, $\frac{1}{2}$, and their conjugates, we can use the fact that complex roots appear in conjugate pairs. So, the roots should include $1 - 3i$ and $\frac{1}{2}$ as well.

Now, let's factor the cubic equation using these roots:

$a (x - (1 + 3i))(x - (1 - 3i))(x - \frac{1}{2}) = 0$

We can expand this and match it to the given form of the equation. Let me proceed with this calculation and give you the values of $a$, $b$, $c$, and $d$.

I'll begin by factoring and expanding this equation. sympy import symbols, expand, conjugate

Define variables

x, a = symbols('x a')

Roots of the equation

root1 = 1 + 3j root2 = 1 - 3j root3 = 1/2

Factor the cubic equation based on the given roots

factorized_form = a * (x - root1) * (x - root2) * (x - root3)

Expand the factored form

expanded_form = expand(factorized_form) expanded_formThe expanded form of the cubic equation, given the roots $1 + 3i$, $1 - 3i$, and $\frac{1}{2}$, is:

$a x^3 - 2.5 a x^2 + 11a x - 5a = 0$

Now, comparing this with the general form $a x^3 + b x^2 + c x + d = 0$, we can equate the coefficients:

• $a = a$
• $b = -2.5a$
• $c = 11a$
• $d = -5a$

From this, we see that the values of $b$, $c$, and $d$ are directly proportional to $a$. To find their numerical values, we would need additional information, such as a specific value for $a$, or another condition related to the equation.

Would you like to proceed with a specific assumption about $a$, or would you like more details on how to solve this further? Let me know if you have any other questions or details you'd like to add.

Here are some related questions you may find interesting:

1. How do we use the properties of complex conjugates to factor polynomials?
2. What happens if the roots of the polynomial are real numbers instead of complex?
3. How can you solve for the coefficients $a$, $b$, $c$, and $d$ if additional information is given?
4. How does the degree of the polynomial affect the number of roots?
5. Can you solve polynomials of higher degrees using similar methods?

Tip: When working with complex roots, always remember that they appear in conjugate pairs, which helps when factoring the polynomial.