Question

Find a quadratic polynomial whose zeroes are $-4$ and $3$ and Verify the relation between zeroes and the coefficients.

Answer 1

Step 1. Form a quadratic polynomial whose zeroes are$-4$ and $3$:

Let the zeroes of quadratic polynomial be $\alpha =-4\text{and}\beta =3$.

$\begin{array}{rcl}{x}^2-(\alpha +\beta )x+\alpha \beta & =& x^2-(-4+3)x+(-4)\left(3\right)\\ & =& x^2+x-12\end{array}$

Therefore, $a=1,b=1,c=-12$

Step 2. Verification:

$\begin{array}{rcl}& \Rightarrow & \alpha +\beta =\frac{-\text{coefficient of}x}{\text{coefficient of}{x}^2}\\ & =& \frac{-b}{a}\\ & =& \frac{-1}{1}\\ & =& -1\\ & \Rightarrow & \alpha +\beta =-4+3\\ & =& -1\end{array}$

$\begin{array}{rcl}& \Rightarrow & \alpha \beta =\frac{\text{constant term}}{\text{coefficient of}{x}^2}\\ & =& \frac{c}{a}\\ & =& \frac{-12}{1}\\ & =& -12\\ & \Rightarrow & \alpha \beta =(-4)\times \left(3\right)\\ & =& -12\end{array}$

Therefore, the relationship between the zeroes and the coefficients is verified.

Hence, $x^2+x-12$ is the required quadratic polynomial.