Question

2. Write a polynomial function of least degree with rational coefficients so that P(x)=0 has the given root.

4 - 5i

Answer 1

p(x) = x^2 - 8x + 41

Explanation:

If a polynomial with rational coefficients has a complex root, then the roots come in complex conjugate pairs. If the polynomial we need to find has the root 4 - 5i, then it must also have its complex conjugate 4 + 5i as a root.

p(x) = [x - (4 - 5i)][x - (4 + 5i)]

p(x) = [(x - 4) + 5i][(x - 4) - 5i]

Now it's a difference of two squares.

p(x) = (x - 4)^2 - (5i)^2

p(x) = x^2 - 8x + 16 + 25

p(x) = x^2 - 8x + 41