Question

Write a quadratic polynomial whose zeroes are $2$ and $ - 6$.

Answer 1

Hint: If zeroes of the quadratic polynomial are given as $\alpha {\text{ and }}\beta $ then the quadratic polynomial can be written as ${{\text{x}}^2} - \left( {\alpha + \beta } \right){\text{x + }}\alpha \beta $ .Put the value of given zeroes in the formula and you’ll get the answer.

Complete step-by-step answer:
Given, the zeroes of a quadratic polynomial are $2$ and $ - 6$.We have to find the quadratic equation. Let $\alpha {\text{ and }}\beta $ be the zeroes of the polynomial then, $\alpha = 2$ and $\beta = - 6$
Now, the quadratic polynomial is written as ${{\text{x}}^2} - \left( {\alpha + \beta } \right){\text{x + }}\alpha \beta $ --- (i)
We have to find the sum of the zeroes and product of the zeroes.
$
   \Rightarrow \alpha + \beta = 2 + \left( { - 6} \right) = - 4 \\
   \Rightarrow \alpha \beta = 2 \times \left( { - 6} \right) = - 12 \\
 $
Since, $\left[
  \left( - \right) \times \left( + \right) = - \\
  \left( - \right) + \left( + \right) = - \\ \right]$
Now on putting the above values in eq. (i), we get
$ \Rightarrow $ ${{\text{x}}^2} - \left( { - 4} \right){\text{x + }}\left( { - 12} \right)$ Since $\left[
  \left( - \right) \times \left( + \right) = - \\
  \left( - \right) + \left( + \right) = - \\ \right]$
On simplifying the equation, we get
$ \Rightarrow {{\text{x}}^2}{\text{ + 4x - 12}}$
Answer- The quadratic equation ${\text{ = }}{{\text{x}}^2}{\text{ + 4x - 12}}$

Note: Zeros of polynomials are the values which when put in the polynomial, make the value of the polynomial equal to zero. Zeroes are also called roots of the polynomial. The quadratic equation has two roots as its degree is also $2$.The quadratic polynomial is written in the form -${{\text{x}}^2} - \left( {{\text{sum of roots}}} \right){\text{x + product of roots}}$.