Which of the following is a polynomial with roots 5, 4i, and -4i?

To find a polynomial with given roots, we can use the fact that if a number is a root of a polynomial, then the polynomial can be expressed in factored form using that root.

In this case, the roots are 5, 4i, and -4i. The corresponding factors of the polynomial will be:

• (x – 5) for the root 5
• (x – 4i) for the root 4i
• (x + 4i) for the root -4i

Now, we can construct the polynomial by multiplying these factors together:

P(x) = (x - 5)(x - 4i)(x + 4i)

Next, we can simplify (x – 4i)(x + 4i) using the difference of squares:

(x - 4i)(x + 4i) = x^2 - (4i)^2 = x^2 + 16

Thus, the polynomial can be rewritten as:

P(x) = (x - 5)(x^2 + 16)

Finally, we can expand this product:

P(x) = x(x^2 + 16) - 5(x^2 + 16) = x^3 + 16x - 5x^2 - 80

This gives us:

P(x) = x^3 - 5x^2 + 16x - 80

So the polynomial with roots 5, 4i, and -4i is:

P(x) = x^3 - 5x^2 + 16x - 80.