Answer :
To factor the polynomial [tex]\(x^3 + x^2 + x + 1\)[/tex] by grouping, we need to follow a step-by-step approach. Here is how it can be done:
1. Group the terms:
We can group the polynomial into two parts to facilitate factoring:
[tex]\[ (x^3 + x^2) + (x + 1) \][/tex]
2. Factor out the common term from each group:
- In the first group, [tex]\(x^3 + x^2\)[/tex], we can factor out [tex]\(x^2\)[/tex]:
[tex]\[ x^2(x + 1) \][/tex]
- In the second group, [tex]\(x + 1\)[/tex], we notice that there is no common factor other than 1, so it remains the same:
[tex]\[ 1(x + 1) \][/tex]
3. Combine the factored groups:
We can see that both groups now have a common factor of [tex]\((x + 1)\)[/tex]:
[tex]\[ x^2(x + 1) + 1(x + 1) \][/tex]
4. Factor out the common binomial factor:
Factor out the [tex]\((x + 1)\)[/tex] from both terms:
[tex]\[ (x + 1)(x^2 + 1) \][/tex]
Thus, the factored form of the polynomial [tex]\(x^3 + x^2 + x + 1\)[/tex] is:
[tex]\[ (x + 1)(x^2 + 1) \][/tex]
The correct answer from the given choices is:
[tex]\[ \left(x^2+1\right)(x+1) \][/tex]
1. Group the terms:
We can group the polynomial into two parts to facilitate factoring:
[tex]\[ (x^3 + x^2) + (x + 1) \][/tex]
2. Factor out the common term from each group:
- In the first group, [tex]\(x^3 + x^2\)[/tex], we can factor out [tex]\(x^2\)[/tex]:
[tex]\[ x^2(x + 1) \][/tex]
- In the second group, [tex]\(x + 1\)[/tex], we notice that there is no common factor other than 1, so it remains the same:
[tex]\[ 1(x + 1) \][/tex]
3. Combine the factored groups:
We can see that both groups now have a common factor of [tex]\((x + 1)\)[/tex]:
[tex]\[ x^2(x + 1) + 1(x + 1) \][/tex]
4. Factor out the common binomial factor:
Factor out the [tex]\((x + 1)\)[/tex] from both terms:
[tex]\[ (x + 1)(x^2 + 1) \][/tex]
Thus, the factored form of the polynomial [tex]\(x^3 + x^2 + x + 1\)[/tex] is:
[tex]\[ (x + 1)(x^2 + 1) \][/tex]
The correct answer from the given choices is:
[tex]\[ \left(x^2+1\right)(x+1) \][/tex]
