Answer:
x³ - 3x² + 81x - 243 = ( x - 3 )( x - 9i )( x + 9i )
Step-by-step explanation:
Given Expression is x³ - 3x² + 81x - 243
We need to factorize the given expression in form of ( x - d )( x - e )( x + f ) where d is a real number, and e and f are complex numbers in the form bi.
From Rational factor theorem,
Factors of 243 is ±1 , ±3 , ±9 , ±27 , ±81 , ±243.
Now by hit and trial we find first factor of real number.
let, p(x) = x³ - 3x² + 81x - 243
put x = 3
p(3) = 3³ - 3 × 3² + 81 × 3 - 243 = 27 - 27 + 243 - 243 = 0
So, The first factor is ( x - 3 ).
Now to find other factor we divide given polynomial with x - 3.
Division is attached.
We get Quotient = x² + 81
Remainder = 0
Now, Factorize x² + 81 = ( x )² - ( 9i )² = ( x - 9i )( x + 9i )
Therefore, x³ - 3x² + 81x - 243 = ( x - 3 )( x - 9i )( x + 9i )
