Let f(x)=x12−x9+x4−x+1 ‌=x9(x3−1)+x(x3−1)+1 ‌=(x3−1)x(x8+1)+1 which is greater than zero for all either x≤0 or x≥1. Again f(x)=x12−x9+x4−x+1=x12−x4(1−x5)+(1−x). which is greater than zero, ∀0<x<1. ∴f(x)>0,∀x∈(−∞,∞)