Question

The polynomial p(x) = x^3 - 19x - 30 has a known factor of (x+2)

Answer 1

We can use synthetic division to divide the polynomial p(x) by (x+2):

-2 | 1 0 -19 -30

| -2 4 -10

|__________________

1 -2 -15 -40

The result is a quadratic polynomial x^2 - 2x - 15 with a remainder of -40.

Therefore, we can factor p(x) as:

p(x) = (x+2)(x^2 - 2x - 15)

The quadratic factor can be factored further as:

p(x) = (x+2)(x-5)(x+3)

So the complete factorization of p(x) is:

p(x) = (x+2)(x-5)(x+3)