Question

One of the factors of x^5+x+1 is. :(i)2x^2-x-1. (2)x^3-x^2+1. (3)x^2-2x^2+1. (4)x^3+x^2+1

Answer 1

Here is ur answer!!

We can solve this problem by undetermined coefficient method.

Please expand this,
➡️ (x^2+ax+b) (x^3+cx^2+dx+e)=

And we get,
➡️ x^5
= +ax^4+cx^4
= +acc^3+bx^3+dx^3
= +adx^2+bcx^2+ex^2
= +eax+bdx
= +eb

Actually those above are already grouped by the similar degree, then we can arrange the simultaneous Eq, by the similar coeff, like this below,
⬇️⬇️⬇️
a+c=0
ac+b+d=0
ad+bc+e=0
ae+bd=1
eb=1

In this case i assume you can solve those Sim. Lin. Eq, five unknown var, and 5 equation.

Then we get,
➡️ a=1, b=1, c=-1, d=0 and e=-1
⏩ (x^2+x+1) (x^3-x^2+1) ➡️ is the factorization

So, the answer is (ii)
x^3-x^2+1 ✔️

Have fun
Faithfully yours,
DivaSept

(If you still didn't understand, you can ask more in the other uploading question)

May be usefull ^_^