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

if -1 is a zero that means x=-1

and x+1 is a factor

divide the cubic by x+1, by either long or synthetic division to get another factor

but that has a quotient of x^2+ 4x -3

x^3 +5x^3 +x -3 = (x+1)(x^2 +4x -3)

there are no integer linear factors of x^2 +4x - 3 but there are linear factors, just irrational ones. Irrational factors come in conjugate pairs.

take x^2 +4x -3 = 0

x^2 +4x = 3

complete the square, add 4 to both sides

x^2 +4x +4 = 3+4

(x+2)^2 = 3+4 = 7

take square roots of both sides

x-2= + or - sqr7

two more zeros are x = 2+sqr7 and x = 2 -sqr7

and the corresponding factors multiplied together are x- one zero times x- the other zero = (x-2-sqr7)(x-2+sqr7)

both are linear factors, but not linear integer factors.

so

(x+1)(x-2-sqr7)(x-2+sqr7) = the original cubic polynomial in terms of linear factor, in a more expansive meaning of "linear factors"