x8−x7−x6+x5+3x4−4x3−2x2+4x−1=0 ⇒x7(x−1)−x5(x−1)+3x3(x−1)−x(x2−1)+2x(1−x)+(x−1)=0 ⇒(x−1)(x7−x5+3x3−x(x+1)−2x+1)=0 ⇒(x−1)(x7−x5+3x3−x2−3x+1)=0 ⇒(x−1)(x5(x2−1)+3x(x2−1)−1(x2−1))=0 ⇒(x−1)(x2−1)(x5+3x−1)=0 ∴x=±1 are roots of above equation and x5+3x−1 is a monotonic term hence vanishs at exactly one value of x other than 1 or −1. ∴3 real roots.