Find a third-degree polynomial equation with rational coefficients that has roots -4 and 3+i.please show work, i am so lost !!

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Since this is a polynomial of degree three, by an application of the fundamental theorem of algebra we know that there are three roots; also all complex roots appear as conjugate pairs.

Thus the three roots are -4,`3+i,3-i`

If `k` is a root, then `(x-k)` is a factor by the factor theorem. Thus the factors of the polynomial are `(x+4),(x-3+i),(x-3-i)`

Then the function is :

`f(x)=a(x+4)(x-3+i)(x-3-i)`

`=a(x+4)[x^2-3x-ix-3x+9+3i+ix-3i-i^2]`

`=a(x+4)(x^2-6x+9-i^2)` But `i^2=-1`

`=a(x+4)(x^2-6x+10)`

`=a(x^3-6x^2+10x+4x^2-24x+40)`

`=a(x^3-2x^2-14x+40)`

You have a choice of any `a` , letting `a=1` we have:

`f(x)=x^3-2x^2-14x+40`

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