Respuesta :

apologiabiology
if it has rational coefients and is a polygon
if a+bi is a root then a-bi is also a root

the roots are -4 and 2+i
so then 2-i must also be a root


if the rots of a poly are r1 and r2 then the factors are
f(x)=(x-r1)(x-r2)

roots are -4 and 2+i and 2-i

f(x)=(x-(-4))(x-(2+i))(x-(2-i))
f(x)=(x+4)(x-2-i)(x-2+i)
expand
f(x)=x³-11x+20

The third-degree polynomial equation with rational coefficients that have roots –4 and 2 + i is  [tex]\rm x^3-11x+20[/tex].

What is the third-degree polynomial?

A third-degree (or degree 3) polynomial is called a cubic polynomial.

[tex]\rm ax^3+bx^2+cx+d[/tex]

A third-degree polynomial equation with rational coefficients that has roots –4 and 2 + i.

The polynomial is;

[tex]\rm (x-(2+i))(x-(2-i))(x-(-4)\\\\ (x+4)(x^2-x(2-i)-x(2+i)+(2+i)(2-i))\\(x+4)(x^2-2x+xi-2x-xi+4-2i+2i-ii^2)\\\\\rm (x+4)(x^2-4x+5)\\\\x(x^2-4x+5) +4(x^2-4x+5)\\\\x^3-4x^2+5x+4x^2-16x+20\\\\x^3-11x+20[/tex]

Hence, the third-degree polynomial equation with rational coefficients that have roots –4 and 2 + i is  [tex]\rm x^3-11x+20[/tex].

Learn more about polynomial here;

https://brainly.com/question/11536910

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