Question

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

Answer 1

Hint: In this question we will use the method of finding the relationship between the zeroes and the coefficient of the polynomials. Here we have given a polynomial and first we will find the zeroes of the polynomial and then by using the method we will verify the relationship between those zeroes and the coefficients.

Complete step-by-step solution -
Given, x$^2$-5
We have , f (x) = x$^2$-5
$ \Rightarrow $ f (x) = (x)$^2$- ($\sqrt 5 {)^2}$
Now, using the identity ( a$^2$- b$^2$) = ( a + b)( a – b), we get
$ \Rightarrow $ f (x) = ( x + $\sqrt 5 $) ( x - $\sqrt 5 $)
The zeroes of f (x) is given by ,
$ \Rightarrow $ f (x) = 0
$ \Rightarrow $ ( x + $\sqrt 5 $) ( x - $\sqrt 5 $) = 0
$ \Rightarrow $ ( x + $\sqrt 5 $) = 0 or, ( x - $\sqrt 5 $) = 0
$ \Rightarrow $ x = $ - \sqrt 5 $ or, x = $\sqrt 5 $.
Hence, the zeroes of the f (x) = x$^2$-5 are : $\alpha $= $ - \sqrt 5 $ and $\beta $ = $\sqrt 5 $.
Now, we have to find the relationship between the zeroes and the coefficient and then we will verify the relationship.
So,
$ \Rightarrow $ Sum of the zeroes = $\alpha $+ $\beta $ = $ - \sqrt 5 $ + $\sqrt 5 $ = 0.
And , $ - \dfrac{{{\text{coefficient of x}}}}{{{\text{coefficient of }}{{\text{x}}^2}}} = - \dfrac{0}{1} = 0$.
We know that, $\alpha $+ $\beta $ = $ - \dfrac{{{\text{coefficient of x}}}}{{{\text{coefficient of }}{{\text{x}}^2}}}$.
$\therefore $ $\alpha $+ $\beta $ = $ - \dfrac{{{\text{coefficient of x}}}}{{{\text{coefficient of }}{{\text{x}}^2}}}$ = 0, [ verified ]
Also,
$ \Rightarrow $ product of zeroes = $\alpha $$\beta $ = ($ - \sqrt 5 $)($\sqrt 5 $) = -5.
And , $\dfrac{{{\text{ constant term}}}}{{{\text{coefficient of }}{{\text{x}}^2}}} = \dfrac{{ - 5}}{1} = - 5$.
We also know that, $\alpha $$\beta $= $\dfrac{{{\text{ constant term}}}}{{{\text{coefficient of }}{{\text{x}}^2}}}$
$\therefore $ $\alpha $$\beta $= $\dfrac{{{\text{ constant term}}}}{{{\text{coefficient of }}{{\text{x}}^2}}}$ = -5 [ verified ]
Hence, we have verified the relationships between the zeroes and the coefficients of the polynomial f (x) = x$^2$-5.

Note : In this type of question, first we have to know the method of finding the zeros of a polynomial. We will find the zeroes of the polynomial and then we will find the relationship between the zeroes ($\alpha $$\beta $ and $\alpha $+ $\beta $ ) and coefficients ( $\dfrac{{{\text{ constant term}}}}{{{\text{coefficient of }}{{\text{x}}^2}}}$, $ - \dfrac{{{\text{coefficient of x}}}}{{{\text{coefficient of }}{{\text{x}}^2}}}$) separately. After that we will compare both the relationships and through this we can easily verify those relationships.