Question

Find the zeroes of the polynomial p(x)=4√3x square -2√3x-2√3 and verify the relationsgip between the coefficient and the zeroes of the polynomial .

Answer 1

Hello,
To find the zeroes we need to factorise the polynomial
Factorization of polynomial by mid - term splitting :-
$4 \sqrt{3} {x}^{2} - 2 \sqrt{3} x - 2 \sqrt{3} \\ = 4 \sqrt{3} {x}^{2} - 4 \sqrt{3} x + 2 \sqrt{3} x - 2 \sqrt{3} \\ = 4 \sqrt{3} x(x - 1) + 2 \sqrt{3} (x - 1) \\ = (x - 1)(4 \sqrt{3} x + 2 \sqrt{3} )$
Now we will equate it with zero
So,
$(x - 1)(4 \sqrt{3} x + 2 \sqrt{3}) = 0 \\ (x - 1) = 0 \\ x = 1 \\ \\ also \\ (4 \sqrt{3} x + 2 \sqrt{3} ) = 0 \\ x = \frac{ - 2 \sqrt{3} }{4 \sqrt{3} } = \frac{ - 1}{2}$
So,
The zeroes of the polynomial are:
1 and - 1/2

Now
$\alpha + \beta = \frac{ - b}{a} \\ 1 + ( - \frac{1}{2} ) = \frac{1}{2} \\ \frac{ - b}{a} = \frac{1}{2 } \\ \\ \alpha \beta = \frac{d}{a} \\ \alpha \beta = 1 \times ( - \frac{1}{2} ) = - \frac{1}{2} \\ \frac{d}{a} = - \frac{1}{2}$
Hence, verified

Hope this will be helping you