Question

How do you factor \[{x^3} + {x^2} + x + 1\] by grouping?

Answer 1

Hint: Here we will first group the first two terms together and the last two terms together. Then we will take common factors from the first two grouped terms and then take common factors from the last two grouped terms. We will simplify it further to get the required answer.

Complete step by step solution:
Here we need to factorize the given polynomial and the given polynomial is \[{x^3} + {x^2} + x + 1\].
Now, we will group the first two terms together and the last two terms together.
\[{x^3} + {x^2} + x + 1 = \left( {{x^3} + {x^2}} \right) + \left( {x + 1} \right)\]
Now, we will take common factors from the first two grouped terms i.e. \[{x^2}\] and then we will take common factors from the last two grouped terms.
\[ \Rightarrow {x^3} + {x^2} + x + 1 = {x^2}\left( {x + 1} \right) + \left( {x + 1} \right)\]
Now, we will take the common factor from the first group and the second group and the factor is \[\left( {x + 1} \right)\].

\[ \Rightarrow {x^3} + {x^2} + x + 1 = \left( {x + 1} \right)\left( {{x^2} + 1} \right)\]

Hence, this is the required factorization of the given polynomial.

Additional information:
The given expression is a cubic expression. When a polynomial has the highest degree of 3 then it is called a cubic polynomial. Similarly, if the expression has the highest degree of 2, then it is called a quadratic polynomial. Also, if the expression has the highest degree of 1, then it is called a linear polynomial. The factor of the given cubic is a combination of linear polynomial and quadratic polynomial.

Note:
Here we have obtained the required factorization of the given polynomial. A polynomial is defined as the algebraic expressions that consist of coefficients and variables. variables are also known as indeterminate. Factorization is a process by which a number is divided into its factors. A factor of any number is defined as the number which exactly divides the given number. Similarly, the factors of the given polynomial are defined as the terms which exactly divide the given polynomials.