√3x^2-8x+4√3=0....
√3x^2-6x-2x+4√3=0....
√3x(x-2√3)-2(x-2√3)=0..
(√3x-2)=0..
(x-2√3)=0..
x=2/√3..
x=2√3 Answer..
zeroes=alpha+beta=-b/a..
2√3+2/√3=8/√3..
8/√3=8/√3..
alpha.beta=c/a..
2√3*2/√3=4√3/√3..
4=4..

Obtaining the Zeros of the Polynomial

To obtain the zeros of the polynomial root 3x^2-8x 4 root 3, we need to set the polynomial equal to zero and solve for x.

3x^2-8x+4root3=0

Using the quadratic formula, we can find the values of x that make the equation true.

x = [8 ± sqrt(64 - 48)]/6

x = [8 ± 4sqrt(2)]/6

x = (4 ± 2sqrt(2))/3

Therefore, the zeros of the polynomial are (4 + 2sqrt(2))/3 and (4 - 2sqrt(2))/3.

Verifying the Relation between Zeros and Coefficients

The relation between the zeros and coefficients of a quadratic equation is given by the following formulas:

- Sum of roots = -b/a
- Product of roots = c/a

where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.

For the given polynomial, a = 3, b = -8, and c = 4root3. Therefore, we have:

- Sum of roots = -(-8)/3 = 8/3
- Product of roots = (4root3)/3

We can verify these values using the zeros we obtained earlier:

- Sum of roots = (4 + 2sqrt(2))/3 + (4 - 2sqrt(2))/3 = 8/3
- Product of roots = [(4 + 2sqrt(2))/3][(4 - 2sqrt(2))/3] = (16 - 8)/9 = 8/9 * 3sqrt(2) = (4root3)/3

Therefore, we have verified that the relation between the zeros and coefficients holds for the given polynomial.