Evaluate the following :

2x4 + 5x3 + 7x2 – x + 41, when x = - 2 - √3i

Given:

⇒ x=-2-i

⇒ 2x⁴+5x³+7x²-x+41

⇒ 2(-2-)⁴+5(-2-i)³+7(-2-i)²-(-2-i)+41

⇒ 2(2⁴+4(2)³(i)+6(2)²(i)²+4(2)(i)³+(i)⁴)-5(2³+3(2)²(i)+3(2)(i)²+(i)³)+7(2²+2(2)(i)+(i)²)+2+i+41

⇒ 16+64i+144i²+48i³+18i⁴-40-60i-90i²-15i³+28+28i+21i²+i+43

We know that i²=-1

⇒ 127+33i+75i²+33i³+18i⁴

⇒ 127+33i+75(-1)+33(-1)(i)+18(-1)²

⇒ 70

∴ 2x⁴+5x³+7x²-x+41=70