Question

Factors of ${x^3} + {x^2} + x + 1$
$\left( A \right)\,\,\left( {x + 1} \right)\left( {{x^2} - 1} \right)\,\,$
$\left( B \right)\,\,\left( {x - 1} \right)\left( {{x^2} + 1} \right)$
$\left( C \right)\,\,\left( {x - 1} \right)\left( {{x^2} - 1} \right)$
$\left( D \right)\,\,\left( {x + 1} \right)\left( {{x^2} + 1} \right)$

Answer 1

Hint: We will do this question by grouping methods. Grouping means that we have to group terms with common factors before factoring. This can be done by grouping a pair of terms and then factor each pair of two terms.

Complete step-by-step solution:
In the given question, we have ${x^3} + {x^2} + x + 1$
The equation shows that the ratio between the first and second terms are the same as that between the third and fourth term.
Then we can say that the method we use here is a factor by grouping to solve a problem.
$ \Rightarrow {x^3} + {x^2} + x + 1$
Now,
Grouping the first two terms with each other and second term with each other.
We get,
\[ \Rightarrow {x^3} + {x^2} + x + 1 = {x^2}\left( {x + 1} \right) + 1\left( {x + 1} \right)\]
\[ \Rightarrow \left( {{x^2} + 1} \right)\left( {x + 1} \right)\]
Hence,
By grouping we get the factor \[\left( {{x^2} + 1} \right)\left( {x + 1} \right)\].
Hence, the correct option is (D).

Additional Information:
(1) Factoring in a polynomial has writing it as a product of two or more polynomials.
(2) In the method of factorisation we reduce it in simple form. The factor can be an integer, a variable or algebra itself in any type of equation.
(3) There are various methods of factorisation. They are factoring out GCF, the grouping method, the difference of square patterns etc.
(4) We have several examples of factoring. However, for this you should take common factors using distributive property.

Note: (1) To use a grouping method we have to group the expression in two groups. As we have two groups, factor out the GCF from the given groups.
(2) If you solve this type of equation the binomial which is in the parenthesis must be the same, if it is not the same then there should be any mistake in the factor.
(3) you should also check the sign properly while writing the final answer.