Question

Find the zeroes of the polynomial ${x^2} - 3$ and verify them.

Answer 1

Hint: To solve this question, we have to know the concept of polynomials and zeros of polynomials. A real number ‘a’ is a zero of polynomials $f\left( x \right)$, if and only if $f\left( x \right) = 0$. Finding a zero of a polynomial $f\left( x \right)$ means solving the polynomial equation for $f\left( x \right) = 0$.

Complete step-by-step answer:
A polynomial is an expression which consists of variables and coefficients, that involves the operation of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.
Let x be a variable, n be a positive integer and as, ${a_1},{a_2},.......,{a_n}$ be constants ( real numbers ). Then,
\[f\left( x \right) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ...... + {a_1}x + {a_0}\] is called a polynomial in variable x.
Given that,
$f\left( x \right) = {x^2} - 3$
To find the zeros of the given polynomial, put $f\left( x \right) = 0$.
$ \Rightarrow f\left( x \right) = {x^2} - 3 = 0 \\
   \Rightarrow f\left( x \right) = {x^2} - {\left( {\sqrt 3 } \right)^2} = 0 $
By using the identity $\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right)$, this can also be written as:
$ \Rightarrow f\left( x \right) = \left( {x + \sqrt 3 } \right)\left( {x - \sqrt 3 } \right) = 0$
We get,
$
  \left( {x + \sqrt 3 } \right) = 0 \\
  x = - \sqrt 3 \\
$ and $
\left( {x - \sqrt 3 } \right) = 0 \\
  x = \sqrt 3 $
Hence, we can say that the zeroes of the polynomial ${x^2} - 3$ are $\sqrt 3 $ and $ - \sqrt 3 $.
Now, we have to verify these zeroes.
So,
$f\left( x \right) = {x^2} - 3$.
To verify the zero ‘a’ of a polynomial $f\left( x \right)$, $f\left( a \right)$ must be 0, i.e. $f\left( a \right) = 0$
Therefore,
$f\left( {\sqrt 3 } \right) = {\left( {\sqrt 3 } \right)^2} - 3 = 3 - 3 = 0$ [ verified ].
$f\left( { - \sqrt 3 } \right) = {\left( { - \sqrt 3 } \right)^2} - 3 = 3 - 3 = 0$ [ verified ].
Hence, the zeroes get verified.

Note: In this type of question, we have to remember some basic points like finding factors of a polynomial, use of identities, and so on. First, we have to find out all the factors of the given polynomial, and then by putting them to 0, we will get the required zeroes. To verify those zeroes, we have to put those zeroes at the place of the variable in the polynomial and then by solving this, if the result obtained is equal to 0, then the zero is verified otherwise not verified.