Question

Find the zeroes of the polynomial $x^2+5x+6$ and verify the relationship between the zeroes and coefficients.

Answer 1

Step 1: Find the zeroes:

Consider ,
$$\begin{gathered} p\left(x\right)=x^2+5x+6 \\ =x^2+3x+2x+6 \\ =x(x+3)+2(x+3) \\ =(x+3)(x+2) \end{gathered}$$
The value of the polynomial $p\left(x\right)=0$
$\begin{array}{cc}\iff & (x+3)(x+2)=0\\ \iff & (x+3)=0\text{or}(x+2)=0\\ \iff & x=-3\text{or}x=-2\end{array}$
So, zeroes of the polynomial $p\left(x\right)=x^2+5x+6$ are $-3$ and $-2$

Step 2: Relationship between zeroes and coefficients:
Take $\alpha =-3$ and $ꞵ=-2$
$\therefore$Sum of the zeroes $=\alpha +ꞵ$

$$\begin{gathered} =(-3)+(-2) \\ =-5 \\ =\frac{\text{Coefficient of}x}{\text{Coefficient of}{x}^2} \end{gathered}$$

Product of the zeroes $=\alpha ꞵ$

$$\begin{gathered} =(-3)x(-2) \\ =6 \\ =\frac{\text{Constant term}}{\text{Coefficient of}{x}^2} \end{gathered}$$
Hence verified