Question

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Answer 1

Answer:

See below.

Step-by-step explanation:

2a.

If a polynomial equation with real coefficients has imaginary roots, then the roots must be pairs of complex conjugate roots.

If -4i is a root, then 4i must be a root, too.

2b.

The roots of a polynomial equation can be written as first degree binomial factors, which when multiplied together will give you the polynomial.

For roots a and b, the first degree binomials are x - a and x - b.

Here the roots are -4i and 4i, so the binomials are x + 4i and x - 4i. Their product will give us the quadratic polynomial.

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

This is the product of a sum and a difference wuich gives the difference of two squares.

= x^2 - (4i)^2

= x^2 - 16(-1)

= x^2 + 16

Answer 2

Least possible degree of polynomial = 5

Step-by-step explanation:

Here we have solutions -5, 1 + 4i, and -4i.

The solutions are 1 real and 2 imaginary.

Real solutions may or may not be with pair.

We know that complex solutions comes with two solutions a + ib and a - ib

So the all solutions of the polynomial are

 -5 , 1+4i, 1-4i, -4i, and 4i

So minimum 5 solutions are there for this polynomial.

Polynomial with 5 solutions are of degree 5.

Least possible degree of polynomial = 5