Question

4. If z = 3 – 4i, then the value of expression z4

– 3z3

+ 3z2 +

99z -95 is equal to ​

Answer 1

Complex Numbers

To remember.

• i = √(- 1)
• i² = - 1
• i³ = - i
• i⁴ = 1

Given. z = 3 - 4i

To find. the value of z⁴ - 3z³ + 3z² + 99z - 95

Solution.

Given, z = 3 - 4i

Then z⁴ = (3 - 4i)⁴

= 3⁴ - 4 (3³) (4i) + 6 (3²) (4i)² - 4 (3¹) (4i)³ + (4i)⁴

= 81 - 4 (27) (4i) + 6 (9) (16i²) - 4 (3) (64i³) + 256i⁴

= 81 - 432i + 864i² - 768i³ + 256i⁴

= 81 - 432i - 864 + 768i + 256

= - 527 + 336i

And z³ = (3 - 4i)³

= 3³ - 3 (3²) (4i) + 3 (3¹) (4i)² - (4i)³

= 27 - 3 (9) (4i) + 4 (3) (16i²) - 64i³

= 27 - 108i + 192i² - 64i³

= 27 - 108i - 192 + 64i

= - 165 - 44i

Also z² = (3 - 4i)²

= 3² - 2 (3) (4i) + (4i)²

= 9 - 24i + 16i²

= 9 - 24i - 16

= - 7 - 24i

∴ z⁴ - 3z³ + 3z² + 99z - 95

= - 527 + 336i - 3 (- 165 - 44i) + 3 (- 7 - 24i) + 99 (3 - 4i) - 95

= - 527 + 336i + 495 + 132i - 21 - 72i + 297 - 396i - 95

= 149

Answer. The value of z⁴ - 3z³ + 3z² + 99z - 95 is 149.