giannisdagoat
Answered

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Factor [tex]\(x^3 + x^2 + x + 1\)[/tex] by grouping. What is the resulting expression?

A. [tex]\((x^2)(x + 1)\)[/tex]
B. [tex]\((x^2 + 1)(x)\)[/tex]
C. [tex]\((x^2 + 1)(x + 1)\)[/tex]
D. [tex]\((x^3 + 1)(x + 1)\)[/tex]

Sagot :

GinnyAnswer
Let's factor the polynomial [tex]\(x^3 + x^2 + x + 1\)[/tex] by grouping. Here are the steps involved in the factorization process:

1. Group the terms in pairs:
[tex]\[ x^3 + x^2 + x + 1 = (x^3 + x^2) + (x + 1) \][/tex]

2. Factor out the common factors in each group:
[tex]\[ = x^2(x + 1) + 1(x + 1) \][/tex]

3. Notice that [tex]\(x + 1\)[/tex] is a common factor in both groups:
[tex]\[ = (x^2 + 1)(x + 1) \][/tex]

The polynomial [tex]\(x^3 + x^2 + x + 1\)[/tex] factors into [tex]\((x^2 + 1)(x + 1)\)[/tex].

Thus, the resulting expression is:

[tex]\[ \boxed{(x^2 + 1)(x + 1)} \][/tex]

So, from the given options, the correct answer is:
[tex]\[ \left(x^2+1\right)(x+1) \][/tex]
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