Question

Factorise
${x}^{5} + x + 1$

Answer 1

we can add and subtract +x²−x²=0without changing the expression:

x^5+x+1=(x^5−x²)+(x²+x+1)=x²(x^3−1)+(x²+x+1)=x²(x−1)(x²+x+1)+(x²+x+1)x^5+x+1=(x^5−x²)+(x²+x+1)=x²(x³−1)+(x²+x+1)=x²(x−1)(x²+x+1)+(x²+x+1)

Now factor out the common factor...which gives us

(x²(x−1)+1)(x²+x+1)=(x³−x²+1)(x²+x+1)

ans = (x³−x²+1)(x²+x+1)