if x=5+2√-4 find the value of x4+9x3+35x2-x+4 - Maths - Complex Numbers and Quadratic Equations

Question

if x=5+2√-4 find the value of
x4+9x3+35x2-x+4

Pronay · Accepted Answer

$x = 5 + 2 \sqrt{- 4} \Rightarrow x = 5 + 4 i$

To find the value of $x^{4} + 9 x^{3} + 35 x^{2} - x + 4$

$x^{2} = (5 + 4 i) (5 + 4 i) = 25 + 40 i - 16 = 9 + 40 i$

$x^{4} = (9 + 40 i) (9 + 40 i) = 81 + 720 i - 1600 = - 1519 + 720 i$

$x^{3} = (9 + 40 i) (5 + 4 i) = 45 + 236 i - 160 = - 115 + 236 i$

Therefore

$x^{4} + 9 x^{3} + 35 x^{2} - x + 4 = (- 1519 + 720 i) + 9 (- 115 + 236 i) + 35 (9 + 40 i) - (5 + 4 i) + 4 = (- 1519 - 1035 + 315 - 1) + i (720 + 2124 + 1400 - 4) = - 2240 + 4240 i$