e
Given that (√2x−i√3) divides P(x) find all roots of P(x)

P(x)=x6−6x5+172x4−7x3+212x2+3xP(x)=x⋅(x5−6x4+172x3−7x2+212x+3)
If (√2x−i√3) divides P(x) then (√2x+i√3) divides P(x)
(√2x−i√3)⋅(√2x+i√3)=(√2x)2−(i√3)2=2x2−(i2⋅3)|i2=−1=2x2−[(−1)⋅3]=2x2+3
{nl} (x5−6x4+172x3−7x2+212x+3):2x2+3=12x3−3x2+72x+1
⇒P(x)=x⋅(√2x−i√3)⋅(√2x+i√3)⋅(12x3−3x2+72x+1)
test x=2: ⇒12x3−3x2+72x+1=0
(12x3−3x2+72x+1):x−2=12x2−2x−12
⇒P(x)=x⋅(√2x−i√3)⋅(√2x+i√3)⋅(x−2)⋅(12x2−2x−12)P(x)=x⋅(√2x−i√3)⋅(√2x+i√3)⋅(x−2)⋅12⋅(x2−4x−1)
Roots:
P(x)=0=x⋅(√2x−i√3)⋅(√2x+i√3)⋅(x−2)⋅12⋅(x2−4x−1)1. Rootx1=02. Root√2x−i√3=0√2x=i√3x2=i√3√23. Root√2x+i√3=0√2x=−i√3x3=−i√3√24. Rootx−2=0x4=25.6. Rootx2−4x−1=0x=4±√42−4⋅(−1)2x=4±√4⋅52x=4±2√52x=2±√5x5=2+√5x6=2−√5
