Question

x^3-3x^2+4x-2=0 if 1-i is a root find x

Answer 1

$\rule{300}3$

$\Huge{\red{\underline{\textsf{Answer}}}}$

The cubic polynomial has zeros :

x = 1, x = 1 - i, x = 1 + i

$\rule{300}2$

$\large{\red{\underline{\tt{Explaination}}}}$

First note that the sum of the coefficients is zero.

That is :

1 - 3 + 4 - 2 = 0

Hence x = 1 is zero and ( x - 1 ) a factor :

x³ - 3x² + 4x - 2 = ( x - 1 ) ( x² - 2x + 2 )

The remaining quadratic factor only has complex zeros, which we can find by completing the square :

a² - b² = ( a - b ) ( a + b )

with a = ( x - 1 ) and b = i as follows

x² - 2x + 2

( x - 1 )² - 1 + 2

( x - 1 )² + 1

[ ( x - 1 ) - i ] [ ( x - 1 ) + i ]

( x - 1 - i ) ( x - 1 + i )

Hence zeros x = 1 $\pm$ i

$\rule{300}2$