To find the zeroes of the polynomial 3x² - x - 4, we need to set the polynomial equal to zero and solve for x:

3x² - x - 4 = 0

We can use the quadratic formula to solve for x:

x = [-(-1) ± √((-1)² - 4(3)(-4))]/(2(3))

x = [1 ± √(49)]/6

x = (1/6)(1 ± 7)

Therefore, the zeroes of the polynomial are:

x = -4/3 and x = 1



The relationship between zeroes and coefficients is given by Vieta's formulas. For a quadratic equation of the form ax² + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a.

For the polynomial 3x² - x - 4, the sum of the roots is:

-(-1)/3 = 1/3

And the product of the roots is:

-4/3 * 1 = -4/3

We can verify that these values match the coefficients of the polynomial. The coefficient of the x term is -1, which is the negative of the sum of the roots. The constant term is -4, which is equal to the product of the roots multiplied by the coefficient of the x² term, which is 3.

Therefore, we have verified that the relationship between the zeroes and coefficients of the polynomial 3x² - x - 4 holds true.