Question

Write a polynomial function P(x) with rational coefficients so that P(x) = 0 has the given roots. 2+√2i, 4+i

Answer 1

uh... ooooof

Explanation:

Answer 2

The polynomial would be p(x) = x⁴ - 12x³ + 55x² -116x + 102

How to write the polynomial with these roots?

We know that the roots must be:

2± √2i and 4 ± i

Then we can write the polynomial as:

p(x) = (x - 2- √2i)*(x - 2+√2i)*(x - 4+i)*(x - 4 - i)

Notice that when x equals any of the roots, the polynomial becomes zero, now just expand the products, we will get:

p(x) = (x - 2- √2i)*(x - 2+√2i)*(x - 4+i)*(x - 4 - i)

p(x) = ((x-2)² + 2 )*( (x - 4)² + 1)

p(x) = (x² - 4x + 4 + 2)*(x² - 8x + 16 + 1)

p(x) = (x² - 4x + 6)*(x² - 8x + 17)

p(x) = x⁴ - 12x³ + 55x² -116x + 102