Question

what is the complete factorization of the polynomial below. x^3-3x^2+x-3

Answer 1

Answer:

(x-3)(x+i)(x-i)

Step-by-step explanation:

$x^3-3x^2+x-3$

We apply grouping method

We group first two terms and last two terms

$(x^3-3x^2)+(x-3)$

Take out GCF x^2 from first group. Also take out 1 from second group

$x^2(x-3)+1(x-3)$

$(x-3) (x^2+1)$

Now we factor x^2 + 1

We  use difference of square formula

a^2 - b^2 = (a+b)(a-b)

x^1 + 1 can be written as $x^2 - (-1)$, the value of -1 is i^2

$x^2 +1 = x^2 -(-1) = x^2 - i^2$

x^2 - i^2 = (x+i)(x-i)

Final answer is (x-3)(x+i)(x-i)