For the polynomial below, is a zero. Express as a product of linear factors.

Question

For the polynomial below, −3

 is a zero.

=g(x)= x^3 + 9x^2 + 20x + 6

Express (gx)

 as a product of linear factors.

Answer 1

By Factor Theorem g(x) = x^3+9x^2+20x +6 has factor x-(-3) = x +3. Let perform synthetic division

-3| 1 9 20 6

-3 -18 -6

1 6 2 0

Second factor of g(x) is x^2 +6x+2. To factor it we solve quadratic equation x^2 +6x +2 = 0

x = -3 + sqrt7 or x = -3 - sqrt7. Because

g(x) = x^3 +9x^2+20x +6 = (x+3)(x+3 - sqrt7)(x+3 + sqrt7)