
Write the zeroes of the polynomial: \[{x^2} + 2x + 1\]
Answer
507k+ views
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.
\[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
\[{\left( {a - b} \right)^2}\; = {a^2} - 2ab + {b^2}\]
\[{a^2} - {b^2} = (a - b)(a + b)\]
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.
\[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
\[{\left( {a - b} \right)^2}\; = {a^2} - 2ab + {b^2}\]
\[{a^2} - {b^2} = (a - b)(a + b)\]
Recently Updated Pages
Master Class 10 English: Engaging Questions & Answers for Success

Master Class 10 Social Science: Engaging Questions & Answers for Success

Master Class 10 Maths: Engaging Questions & Answers for Success

Master Class 10 Science: Engaging Questions & Answers for Success

Class 10 Question and Answer - Your Ultimate Solutions Guide

Master Class 11 Economics: Engaging Questions & Answers for Success

Trending doubts
Why is there a time difference of about 5 hours between class 10 social science CBSE

When and how did Canada eventually gain its independence class 10 social science CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Write examples of herbivores carnivores and omnivo class 10 biology CBSE

10 examples of evaporation in daily life with explanations
