please show work thank you A polynomial P is given Px x3 3x2

please show work thank you

A polynomial P is given P(x) x3 3x2 4x 12 (a) Factor P into linear and irreducible quadratic factors with real coefficients P(x) (b) Factor P completely into linear factors with complex coefficients P(x)

Solution

P(x) = x^3 -3x^2 +4x -12

a) quadratic factors   P(x) = (x^3 -3x^2) + (4x -12)

x^2( x-3) +4(x-3)

P(x) = (x^2+4)(x-3)

b) Linear factors :

P(x) = (x^2+4)(x-3)

x^2 +4 = (x+2i)(x-2i)

So, P(x) = (x^2+4)(x-3) =  (x+2i)(x-2i)( x-3)