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Use the grouping method to factor the polynomial below completely.

[tex]\[ x^3 + 2x^2 + 3x + 6 \][/tex]

A. [tex]\(\left(x^2 + 2\right)(x + 3)\)[/tex]

B. [tex]\(\left(x^2 + 2\right)(x + 2)\)[/tex]

C. [tex]\(\left(x^2 + 3\right)(x + 3)\)[/tex]

D. [tex]\(\left(x^2 + 3\right)(x + 2)\)[/tex]

Sagot :

GinnyAnswer
To factor the polynomial [tex]\( x^3 + 2x^2 + 3x + 6 \)[/tex] completely using the grouping method, follow these detailed steps:

1. Group the terms: Split the polynomial into two groups.
[tex]\[ x^3 + 2x^2 + 3x + 6 = (x^3 + 2x^2) + (3x + 6) \][/tex]

2. Factor out the greatest common factor (GCF) in each group:
- In the first group, [tex]\( x^3 + 2x^2 \)[/tex], the GCF is [tex]\( x^2 \)[/tex].
- In the second group, [tex]\( 3x + 6 \)[/tex], the GCF is [tex]\( 3 \)[/tex].

Factoring out the GCF from each group:
[tex]\[ x^2(x + 2) + 3(x + 2) \][/tex]

3. Factor out the common binomial factor: Notice that both terms have a common factor of [tex]\( (x + 2) \)[/tex].
[tex]\[ x^2(x + 2) + 3(x + 2) = (x + 2)(x^2 + 3) \][/tex]

Thus, the factored form of the polynomial [tex]\( x^3 + 2x^2 + 3x + 6 \)[/tex] is:
[tex]\[ (x + 2)(x^2 + 3) \][/tex]

Therefore, the correct answer is:

D. [tex]\( (x^2 + 3)(x + 2) \)[/tex]
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