Question:
If $\alpha, \beta$ are the zeros of the polynomial $2 y^{2}+7 y+5$, write the value of $\alpha+\beta+\alpha \beta$.
Solution:
Let $\alpha$ and $\beta$ are the zeros of the polynomial $f(x)=2 y^{2}+7 y+5$. Then
The sum of the zeros $=\frac{-\text { Coefficient of } y}{\text { Coefficient of } y^{2}}=\frac{-7}{2}$ The product of the zeros $=\frac{\text { Constant term }}{\text { Co-efficient of } y^{2}}=\frac{5}{2}$
Then the value of $\alpha+\beta+\alpha \beta$ is
$\alpha+\beta+\alpha \beta$
$=\frac{-7}{2}+\frac{5}{2}$
$=\frac{-7+5}{2}$
$=\frac{-2}{2}$
$=-1$
Hence, the value of $\alpha+\beta+\alpha \beta$ is $-1$