Question

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

Answer 1

x³-5x²+16x-80 = 0

Explanation:

We have given a set of zeros.

5,4i and -4i

To make a polynomial with a set of given zeros, we can use the fact that a is zero of polynomial if and only if (x-a) is a factor of the polynomial.Then starting from given zeros,we will take the product of factors.

(x-5)(x-4i)(x-(-4i)) = 0

(x-5)(x-4i)(x+4i) = 0

(x-5)(x²-16i²) = 0

(x-5)(x²-16(-1)) = 0 ∵ i² = -1

(x-5)(x²+16) = 0

x³+16x-5x²-80 = 0

x³-5x²+16x-80 = 0 is the polynomial with roots 5,4i and -4i.

Answer 2

`a(x^3-5x^2+16x-80)`, for any real a.

Explanation:

The three roots of a polynomial is given

Hence the polynomial would be cubic having factors as

x-5, x+4i and x-4i

Multiplying we get

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

This would be the possible least degree polynomial with the given roots.

Even if we multiply this by any real number, roots would remain the same

Hence cubic polynomial is

`a(x^3-5x^2+16x-80)`

for some real a.