What polynomial has roots 5, 4i, and -4i?

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To find a polynomial with specific roots, we can construct it using the fact that if a number r is a root, then the factor (x – r) is part of the polynomial.

In this case, the given roots are:

  • 5
  • 4i
  • -4i

Starting with the root 5, the corresponding factor is:

(x – 5)

Now, for the complex roots, 4i and -4i, we find their corresponding factor:

(x – 4i)(x + 4i)

Using the difference of squares formula, we can simplify this factor:

(x – 4i)(x + 4i) = x^2 – (4i)^2 = x^2 – 16(-1) = x^2 + 16

Now, we multiply the factors together:

(x – 5)(x^2 + 16)

Next, we distribute:

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

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

x³ – 5x² + 16x – 80

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