College algebra

Question

can you write a polynomial that satisfies the given condition?

degree 3 polynomial with integer coefficients with zeros -5i and 7/5

Answer 1

If -5i is a root, then so is 5i.

That means (x - 5i) and (x + 5i) are factors of the function.

If 7/5 is a root, then (x - 7/5) is a factor. This can also be written as (5x - 7).

So, p(x) = (x - 5i) ( x + 5i) (5x -7)

(x - 5i) ( x + 5i) when multiplied results in a difference of squares: (x² - 25i²) = (x² + 25).

So,

p(x) = (x² + 25) (5x - 7) = 5x³ - 7x² + 125x -175

Note that there are infinitely many possible answers because if the function is written as follows the results are the same (where A is any nonzero integer).

p(x) = A (5x³ - 7x² + 125x - 175)

Check it out here:

desmos.com/calculator/dzvzpb32iv