The function f(x)=x^4−15(x^3)+81(x^2)−201x+182 has four complex roots, one of which is 3−2i. What is the sum of all real and imaginary coefficients of these roots?
Details and assumptions
i is the imaginary unit, where i2=−1.
one other complex root must be 3+2i
So, (x-(3-2i))(x-(3+2i)) are factors of f(x)
That is, (x-3)^2+4 = (x^2-6x+13) divides f(x)
f(x) = (x^2-6x+13)(x^2-9x+14)
= (x^2-6x+13)(x-2)(x-7)
So, the roots are
2,7,3-2i,3+2i
You can add 'em up
To find the sum of all real and imaginary coefficients of the roots of the given function, we first need to find all the roots. Let's call the roots of the function as r1, r2, r3, and r4.
Given that one of the roots is 3 - 2i, we know that the complex roots of a polynomial come in complex conjugate pairs. Therefore, the other complex root must be 3 + 2i.
To find the other roots, we can use synthetic division or any other root-finding method. However, in this case, since we're only interested in the sum of coefficients, we can instead use Vieta's formulas.
Vieta's formulas state that for a polynomial of the form ax^n + bx^(n-1) + ... + k = 0, the sum of the roots is given by the negation of the coefficient of the term one degree less than the highest power (n-1) divided by the coefficient of the highest power term (a).
In our case, the polynomial is f(x) = x^4 - 15x^3 + 81x^2 - 201x + 182. Therefore, the sum of the roots is -(-15)/1 = 15.
Since we know two of the roots are complex conjugates, let's call the other two roots r3 and r4.
The sum of all the roots of a polynomial is equal to the negation of the coefficient of the term one degree less than the highest power (n-1) divided by the coefficient of the highest power term (a). In this case, since the highest power is 4 and the coefficient is 1, the sum of all the roots is -(-15)/1 = 15.
Since the complex roots come in complex conjugate pairs, their sum will be equal to 2 * Re(ComplexRoot), where Re(ComplexRoot) is the real part of the complex root.
In our case, the sum of the complex roots is (3 - 2i) + (3 + 2i) = 6.
Now we can find the sum of all the real and imaginary coefficients of the roots:
Real coefficients:
Since the coefficients of polynomial f(x) are all real, the sum of real coefficients is 0.
Imaginary coefficients:
The imaginary coefficients are obtained from the complex roots. In this case, the sum of the imaginary coefficients is the sum of the imaginary parts of the complex roots, which is -2i + 2i = 0.
Therefore, the sum of all real and imaginary coefficients of the roots is 0.