Respuesta :

apologiabiology
for a polynomial with ratinal coefients, if a+bi is a root, then a-bi is a root
since 0+3i is a root, 0-3i is also a root

if the roots of an equtation are r1,r2, r3, the factored form of the polynomial is like this
(x-r1)(x-r2)(x-r3)

given
roots are -5, 3i and -3i
(x-(-5))(x-3i)(x-(-3i)) or
(x+5)(x-3i)(x+3i)
if we expand
x³+5x²+9x+45
aksnkj

The polynomial with rational coefficients and roots -5 and 3i will be [tex]p(x)=x^3+5x^2+9x+45=0[/tex].

Given information:

Polynomial p(x)=0 has the roots: -5 and 3i

It is required to write a polynomial with rational coefficient and the given roots.

The polynomial has a complex root 0+3i. So, there will be one more root of the polynomial which is 0-3i.

Now, the roots of the polynomial are

[tex]\alpha =-5\\\beta=0+3i\\\gamma=0-3i[/tex]

So, the required polynomial can be written as,

[tex](x-\alpha)(x-\beta)(x-\gamma)=(x-(-5))(x-(0+3i))(x-(0-3i))\\=(x+5)(x-3i)(x+3i)\\=(x+5)(x^2-(3i)^2)\\=(x+5)(x^2+9)\\=x(x^2+9)+5(x^2+9)\\=x^3+5x^2+9x+45[/tex]

Therefore, the polynomial with rational coefficients and roots -5 and 3i will be [tex]p(x)=x^3+5x^2+9x+45=0[/tex].

For more details about polynomial, refer to the link:

https://brainly.com/question/17822016

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