CBSE · Class 10 · Mathematics · PolynomialsExplain the relationship between the zeros and the coefficients of a quadratic polynomial $ax^2 + bx + c$.

For a quadratic polynomial $p(x) = ax^2 + bx + c$, where $a \neq 0$, let $\alpha$ and $\beta$ be its two zeros. The relationship between these zeros and the coefficients is defined by two main formulas. Firstly, the sum of the zeros is equal to the negative ratio of the coefficient of $x$ to the coefficient of $ax^2$, expressed as $\alpha + \beta = -b/a$. Secondly, the product of the zeros is equal to the ratio of the constant term to the coefficient of $ax^2$, expressed as $\alpha \beta = c/a$. These relationships are essential for verifying the roots of quadratic equations.