The function
f(x)=x^4−10x^3+40x^2−80x+64
has four complex roots, one of which is 2−2i. What is the sum of all real and imaginary coefficients of these roots?
i here is imaginary unit i.e. i^2 = -1
It's a simple question and i hope that you should try a bit to solve it
although the answer is 10
it's a brilliant question ??
I got it its roots are 4 , 2, 2-2i, 2+2i
To find the sum of all real and imaginary coefficients of the roots, we need to determine the other three complex roots of the function f(x). We know that one of the complex roots is given as 2 - 2i, so we can use this information to find the other roots.
Let's first find the complex conjugate of 2 - 2i. The complex conjugate of a complex number a + bi is obtained by changing the sign of the imaginary part, so the complex conjugate of 2 - 2i is 2 + 2i.
Now, we can use the fact that a polynomial with real coefficients has roots in complex conjugate pairs. This means that if 2 - 2i is a root, then 2 + 2i must also be a root.
Since we have found two roots so far (2 - 2i and 2 + 2i), we need to find the remaining two roots. To find these roots, we can divide the original polynomial by the factors corresponding to the known roots.
Using polynomial long division or synthetic division, divide the function f(x) by (x - 2 + 2i) and (x - 2 - 2i) to obtain a quadratic quotient. The quadratic quotient will represent the remaining two complex roots.
Once you have obtained the quadratic quotient, you can factor it or use the quadratic formula to solve for the remaining complex roots.
Finally, sum up all the real and imaginary coefficients of these roots (including the known roots) to get the desired result.
This process involves some manual calculations, so let me know if you need help with the specific calculations or any further clarification.