Question

What is the factorization of a quadratic polynomial?

Answer 1

Factorization of quadratic Polynomial:

A polynomial of degree $2$ is called a quadratic polynomial.

It is of the form $p\left(x\right)=a{x}^2+bx+c$, where $a\ne 0$.

Factorize the quadratic equation $p\left(x\right)=a{x}^2+bx+c$:

Take $p$ and $q$ such that $p+q=b$ and $pq=ac$,

hence,

$$\begin{gathered} a{x}^2+bx+c=a{x}^2+\left(p+q\right)x+\frac{pq}{a} \\ \Rightarrow a{x}^2+bx+c=a{x}^2+px+qx+\frac{pq}{a}\left[\because pq=ac\right] \\ \Rightarrow a{x}^2+bx+c=x\left(ax+p\right)+\frac{q}{a}\left(ax+p\right) \\ \Rightarrow a{x}^2+bx+c=\left(ax+p\right)\left(x+\frac{q}{a}\right) \end{gathered}$$

Hence, the factorization of the quadratic polynomial is $\left(ax+p\right)\left(x+\frac{q}{a}\right)$.