Find the polynomial of lowest degree with only real coefficients and having the given zeros. - 7i and square root of 2

x³ - (√2) x² + 49x - 49√2

Step-by-step explanation:

If one root is - 7i, another root must be 7i. You can't just have one root with i. The other roos is √2, so there are 3 roots.

x = - 7i is one root,

(x + 7i) = 0 is the factor

x = 7i is one root

(x - 7i) = 0 is the factor

x = √2 is one root

(x - √2) = 0 is the factor

So the factors are ...

(x + 7i) (x - 7i) (x - √2) = 0

Multiply these out to find the polynomial ...

(x + 7i) (x - 7i) = x² + 7i - 7i - 49i²

Which simplifies to

x² - 49i² since i² = - 1, we have

x² - 49 (-1)

x² + 49

Now we have ...

(x² + 49) (x - √2) = 0

Now foil this out ...

x² (x) - x² (-√2) + 49 (x) + 49 (-√2) = 0

x³ + (√2) x² + 49x - 49√2