Obtain all zeros of the polynomial f(x) = 2x⁴ + x³ – 14x² –19x – 6, if two of its zeros are –2 and –1.

We know that if (x – a) is a zero of a polynomial then (x – a) is a factor of f(x)

Since –2 and –1 are zeros of f(x).Therefore, (x + 2)(x + 1) = x² + 3x + 2 is a factor of f(x).

Now on dividing f(x) = 2x⁴ + x³ – 14x² –19x – 6 by g(x) = x² + 3x + 2 to find other zeros.

By applying division algorithm, we have:

2x⁴ + x³ – 14x² –19x – 6 = (x² + 3x + 2)(2x² – 5x – 3)

2x⁴ + x³ – 14x² –19x – 6 = (x + 2)(x + 1)(2x² – 5x – 3)

2x⁴ + x³ – 14x² –19x – 6 = (x + 2)(x + 1)(2x + 1)(x – 3)

Hence, the zeros of the given polynomial are –1/2, –1, –2, 3.