Question

Write the zeroes of the polynomial: \[{x^2} + 2x + 1\]

Answer 1

Hint:
Here, we will use the concept of the factorisation. Factorisation is the process in which a number is written in the forms of its small factors which on multiplication give the original number. Firstly we will split the middle term of the equation and then we will form the factors. Then we will equate these factors to zero to get the value of \[x\] which are the zeroes of the polynomial.

Complete step by step solution:
The polynomial equation is \[{x^2} + 2x + 1\].
Firstly we will split the middle term into two parts such that its multiplication will be equal to the product of the first term and the third term of the equation. Therefore, we get
\[ \Rightarrow {x^2} + x + x + 1\]
Now we will be taking \[x\] common from the first two terms and taking 1 common from the last two terms. Therefore the equation becomes
\[ \Rightarrow x\left( {x + 1} \right) + 1\left( {x + 1} \right)\]
Now we will take \[\left( {x + 1} \right)\] common from the equation we get
\[ \Rightarrow \left( {x + 1} \right)\left( {x + 1} \right)\]
Therefore, these are the factors of the equation. Now we will equate these factors to zero to get the value of \[x\]. Therefore, we get
\[ \Rightarrow x + 1 = 0\]
\[ \Rightarrow x = - 1\]
As both the factors are the same therefore the value of \[x\] is \[ - 1, - 1\].

Hence, \[ - 1, - 1\] are the zeroes of the given polynomial.

Note:
Here we will note that zeroes are that value of the equation at which the value of the equation becomes zero. We should split the middle term very carefully according to the basic condition which is that the middle term i.e. term with the single power of the variable should be divided in such a way that its multiplication must be equal to the product of the first and the last term of the equation. Factors can be the same and I can be different. We should know that the factors we have obtained on solving that we will get the original given equation. Factors are the smallest part of the number or equation which on multiplication will give us the actual number of equations. Generally in these types of questions algebraic identities are used to solve and make the factors. It is very important that we learn about algebraic identities in maths.
\[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
\[{\left( {a - b} \right)^2}\; = {a^2} - 2ab + {b^2}\]
\[{a^2} - {b^2} = (a - b)(a + b)\]