Question

Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros.
1 + i, 3

Answer 1

Answer:

$f(x) = x^{3} - 5x^{2}   + 7x - 3$

Step-by-step explanation:

If the polynomial has one root as 1 + i, it's conjugate 1 - i will also be its root.

Therefore, the polynomial has roots 1 + i, 1 - i, and 3.

So, the polynomial function of lowest degree with rational coefficients will be  

f (x) = (x - 3)(x - 1 - i)(x - 1 + i)  

⇒ $f(x) = (x - 3)[ x^{2} - 2x + (1 + i)(1 - i)]$

⇒$f(x) = (x - 3)(x^{2} - 2x + 2)$

⇒ $f(x) = x^{3} - 5x^{2}   + 7x - 3$ (Answer)