Question

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

Answer 1

Answer:

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

Step-by-step 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.

Answer 2

Answer:

x³-5x²+16x-80  = 0

Step-by-step 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.