Question

iv) 2x3 - 11x2+44x+27, if x =25/ 3-4i; .​

Answer 1

Answer:

2x³ - 11x²+ 44x + 27 = 2

Step-by-step explanation:

Since

x = 25 / (3 - 4i)

By Rationalization

x = 25 ( 3 + 4i ) / ( 3² - (4i)² )

x = 25 ( 3 + 4i ) / ( 9 + 16 ) = 25( 3 + 4i )/25 = 3 + 4i

So

x = 3 + 4i

Now using (a + b)³ = a³ + 3ab(a + b) + b³  

x³ = ( 3 + 4i )³ = (3)³ + 3(3)(4i)(( 3 + 4i ) + (4i)³

   = 27 + 36i( 3 + 4i ) - 64i = 27 + 108i - 144 - 64i = -117 + 44i

   = -117 + 44i

⇒ 2x³ = 2(-117 + 44i) = -234 + 88i

And using (a + b)² = a² + 2ab + b²

x² = ( 3 + 4i )² = 9 - 16 + 24i = -7 + 24i

⇒ 11x² = 11(-7 +24i) = -77 + 264i

And

44x = 44( 3 + 4i ) = 132 + 176i

Putting all the values we get

2x³ - 11x²+ 44x + 27 = -234 + 88i - (-77 + 264i) + 132 + 176i + 27

                                 = - 234 + 77 + 132 + 27 + (88 - 264 + 176)i = 2 + 0i

Hence

2x³ - 11x²+ 44x + 27 = 2