Respuesta :
We are given polynomial function f(x)=[tex]x^4-x^3-19x^2-x-20[/tex]
We are given two zeros -4 and 5.
Therefore, x=-4 and x=5 is given.
So, the two factors of the polynomial will be (x+4)(x-5).
Let us write some steps to factor the given polynomial.
Let us take above factor (x+4) first.
On factoring (x+4) we get factors
[tex]x^4-x^3-19x^2-x-20=\left(x+4\right)\left(x^3-5x^2+x-5\right)[/tex]
[tex]\mathrm{:Let \ us \ factor}\:x^3-5x^2+x-5 \ now.[/tex]
Grouping
[tex]=\left(x^3-5x^2\right)+\left(x-5\right)[/tex]
Factoring out gcf from each group, we get
[tex]=x^2\left(x-5\right)+\left(x-5\right)[/tex]
[tex]=\left(x-5\right)\left(x^2+1\right)[/tex]
So, the final factored form of given polynomial will be
[tex]x^4-x^3-19x^2-x-20=\left(x+4\right)\left(x-5\right)\left(x^2+1\right)[/tex]
For first to factors (x+4) and (x-5) we have given roots: -4 and 5.
Let us find the root of third factor we got.
[tex]x^2+1=0[/tex]
Subtracting both sides by 1.
x^2 = - 1.
On taking square root on both sides, we get a square root(-1)
[tex]x=\sqrt{-1}=[/tex]+ i and -i.
Those are imaginary solutions.
Therefore, correct option is B) No, there are two imaginary solutions.
