for the polynomial below, -1 is a zero. f(x)=x^3+5x^2+x-3 express f(x) as a product of linear factors

Question

for the polynomial below, -1 is a zero. f(x)=x³+5x²+x-3 express f(x) as a product of linear factors

Answer 1

one factor is x+1

use long or sythetic division to divide the cubic by x+1 to get x^2 + 4x -3

that doesn't factor further into linear factors

f(x) = (x+1)(x^2 +4x -3)

Possibly the problem was mis-copied or had a mistake in it from the beginning.

Best you might do is divide by x+1 again and get:

f(x) = (x+1)(x+1)[x+3 - 6/(x+1)] but that last factor is not linear

or you could use the quadratic formula on the quadratic factor to get

x = -4/2 + or - (1/2)sqr(16+12) =-1/2 + or - (1/2)2sqr7

x = -2 + or - sqr7

x = -2 +sqr7 or -2 -sqr7

then there're two linear factors

(x+2-sqr7) and (x+2+sqr7)

the cubic f(x) = (x+1)(x+2-sqr7)(x+2+sqr7)

= about (x+1)(x+4.65)(x-0.65)

then they are linear factors but not rational factors. Multiply them out and you get the original cubic polynomial