The function f(x)=x4−15x3+81x2−201x+182 has four complex roots, one of which is 3−2i. What is the sum of all real and imaginary coefficients of these roots?
15
15................apply vieta's rule . :))))))))
Elaborate the Answer !!! pls
To find the sum of all real and imaginary coefficients of the roots, we need to consider the roots of the equation. Since one of the complex roots is given as 3-2i, we know that the corresponding complex conjugate, 3+2i, is also a root. Complex roots always come in conjugate pairs when the coefficients of the polynomial equation have real values.
Given that the polynomial equation has four complex roots, we have found two of them: 3-2i and 3+2i. To find the remaining complex roots, we can use the fact that the sum of all roots of a polynomial is equal to the opposite of the coefficient of the second-to-last term divided by the coefficient of the leading term. In this case, the second-to-last term is -201x, and the leading term is x^4. Therefore, the sum of all the roots will be -(-201) / 1 = 201.
Since we already have two complex roots, the remaining two roots must also be complex and form a conjugate pair. Let's call these roots a+bi and a-bi. To find their sum, we can add them up: (3-2i) + (3+2i) + (a+bi) + (a-bi).
Since the sum of all roots is 201, we can simplify the equation: 201 = 6 + 2a.
By rearranging the equation, we find that 2a = 195, and solving for a, we get a = 97.5.
To find the sum of all the roots, we substitute the value of a into the equation: 6 + 2(97.5) = 201.
Therefore, the sum of all real and imaginary coefficients of these roots is 201.