The given polynomial is ⇒⇒⇒ f(x) = x³ - 3x² + 81x - 243
by factoring the absolute term (243) to find one of the factors of the the polynomial
∴ 243 = (1 * 243) or (3 * 81) or (9 * 27)
check which of the numbers {1 , 3 , 9 , 27 , 81 , 243} make f(x)= 0
i have checked 3 and it makes polynomial = 0
i.e: f(3) = 0 ⇒⇒ (x - 3) is one of the factors of f(x)
By using the reminder theorem ⇒⇒ see the attache figure
∴ [tex] \frac{f(x)}{x-3} = x^{2} +81 [/tex]
And ⇒⇒ (x² + 81) is a sum of two squares which can be factored using the complex numbers as following
x² + 81 = ( x + 9i ) ( x - 9i )
∴ f(x) = x³ - 3x² + 81x - 243 = (x - 3)(x + 9i)(x - 9i)
