Question:
Evaluate zⴠ+9z³ +3z²-z+ 4,where z =-5+ 2√-4
Answer:
Given z = -5 + 2√(-4)
=> z = -5 + 2√(-1 * 4)
=> z = -5 + 2 * √(-1) * √4
=> z = -5 + 4 * i {since √(-1) = i}
=> z = -5 + 4i
=> z = 4i - 5
Now, z4 + 9z3 + 3z2 - z + 4
= (4i - 5)4 + 9(4i - 5)3 + 3(4i - 5)2 - (4i - 5) + 4
= (4i - 5)2 * (4i - 5)2 + 9(4i - 5) * (4i - 5)2 + 3(4i - 5)2 - (4i - 5) + 4
= (16i2 + 25 - 40i) * (16i2 + 25 - 40i) + 9(4i - 5) * (16i2 + 25 - 40i) + 3(16i2 + 25 - 40i) - (4i - 5) + 4
= (-16 + 25 - 40i) * (-16 + 25 - 40i) + 9(4i - 5) * (-16 + 25 - 40i) + 3(-16 + 25 - 40i) - (4i - 5) + 4
= (9 - 40i) * (9 - 40i) + 9(4i - 5) * (9 - 40i) + 3(9 - 40i) - (4i - 5) + 4
= (9 - 40i){9 - 40i + 9(4i - 5) + 3} - (4i - 5) + 4
= (9 - 40i){9 - 40i + 36i - 45 + 3} - (4i - 5) + 4
= (9 - 40i){-33 - 4i} - 4i + 5 + 4
= -297 - 36i + 1320i + 160i2 - 4i + 9
= -297 - 36i + 1320i - 160 - 4i + 9
= -448 + 1280i
So, z4 + 9z3 + 3z2 - z + 4 = -448 + 1280i
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