Question

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

Answer 1

Determine the quadratic polynomial.

Quadratic equations or quadratic polynomials are second-degree algebraic expressions and are of form $a{x}^2+bx+c=0$.

The standard form of a quadratic polynomial with roots $m$ and $n$is $x^2–(m+n)x+mn=0.$

Given roots are $2,-6$

$$\begin{gathered} \therefore m=2 \\ n=-6 \end{gathered}$$

Calculate the sum of the roots:

$$\begin{gathered} m+n=2+(-6) \\ \Rightarrow =-4 \end{gathered}$$

Calculate the product of the roots:

$$\begin{gathered} m\times n=2\times (-6) \\ \Rightarrow =-12 \end{gathered}$$

Putting the sum and product root value into the standard form of a quadratic polynomial:

$$\begin{gathered} x^2–(m+n)x+mn=0 \\ \Rightarrow x^2–(-4)x+(-12)=0 \\ \Rightarrow x^2+4x-12=0 \end{gathered}$$

Hence, the quadratic polynomial is $x^2+4x-12=0.$