Question

Find the zeros of the following quadratic polynomials $x^2-1$ and verify the relationship between the zeros and the coefficients.

Answer 1

Step 1: Form the equation

The given polynomial is $x^2-1=x^2+0.x-1$.

We know that the zeroes of a polynomial are evaluated by equating them with zero.

$\therefore p\left(x\right)=0$

$\Rightarrow x^2-1=0$

Step 2: Solve to find the zeroes

$x^2-1=0$

$\Rightarrow$ $x^2-1^2=0$

$\Rightarrow$$\left(x+1\right)\left(x-1\right)=0$ $\left[\because a^2-b^2=\left(a+b\right)\left(a-b\right)\right]$

$\Rightarrow$ $x=-1,1$

Step 3: Verification

We know that for a given polynomial $a{x}^2+bx+c$,

Sum of the zeroes$=-\frac{b}{a}$ and product of the roots$=\frac{c}{a}$

Sum of the zeroes $=-1+1=0$

Again, $-\frac{b}{a}=-\frac{0}{1}=0$

Product of the zeroes $=\left(-1\right)1=-1$

Again, $\frac{c}{a}=\frac{-1}{1}=-1$

Thus, the relationship between the zeroes and the coefficients of the polynomial is verified.

Hence, the zeroes of this polynomial are $-1$ and $1$.