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

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

+a x^4 + c x^4

+a c x^3 + b x^3 + d x^3

+a d x^2 + + b c x^2 +e x^2

+e a x + b d x

+e b

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.

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

then we get:

a=1, b=1, c=-1, d=0 dan 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,

Jupe01