Question

# Find the quadratic polynomial whose zeroes are given as -3, 5. Choose the correct option:A. ${{x}^{2}}-3x-15$B. ${{x}^{2}}-2x-7$C. ${{x}^{2}}-2x-15$D. ${{x}^{2}}-2x+15$

Hint: The quadratic equation with the roots $\alpha ,\beta$ is given by $(x-\alpha )(x-\beta )=0$.
Now we know that if we have roots $\alpha ,\beta$, then the quadratic equation will be $(x-\alpha )(x-\beta )=0$ and the quadratic polynomial will be $(x-\alpha )(x-\beta )$.
Now, here we are given with the roots $\alpha =-3,\,\,\beta =5$. So, the quadratic equation will be, as follows:
\begin{align} & \Rightarrow (x-\alpha )(x-\beta )=0 \\ & \Rightarrow (x+3)(x-5)=0 \\ & \Rightarrow x\times x-5x+3x-3\times \;5=0 \\ & \Rightarrow {{x}^{2}}-2x-15=0 \\ \end{align}
So now the required quadratic polynomial will be ${{x}^{2}}-2x-15$.
Hence the correct answer is option (C) ${{x}^{2}}-2x-15$.
Note: There is a difference between the equation and the polynomial. The equation is $(x-\alpha )(x-\beta )=0$ , but the polynomial is just $(x-\alpha )(x-\beta )$.