Factor completely: [tex]2x^3 + 14x^2 + 6x + 42[/tex]

A. [tex]2\left[\left(x^2+3\right)(x+7)\right][/tex]
B. [tex]\left(2x^2+6\right)(x+7)[/tex]
C. [tex]\left(x^2+3\right)(2x+14)[/tex]
D. [tex]2\left(x^3+7x^2+3x+21\right)[/tex]



Answer :

To factor the polynomial [tex]\( 2x^3 + 14x^2 + 6x + 42 \)[/tex] completely, follow these steps:

1. Identify Common Factors:
Examine each term in the polynomial to see if there is a common factor. Here, all terms are divisible by 2.

[tex]\[ 2x^3 + 14x^2 + 6x + 42 = 2(x^3 + 7x^2 + 3x + 21) \][/tex]

We factor out the 2:

[tex]\[ = 2 (x^3 + 7x^2 + 3x + 21) \][/tex]

2. Factor the Polynomial Inside the Parentheses:
Next, focus on [tex]\( x^3 + 7x^2 + 3x + 21 \)[/tex]. We need to factor this polynomial further.

3. Group Terms:
Try to group terms that can be factored easily.

[tex]\[ x^3 + 7x^2 + 3x + 21 = (x^3 + 7x^2) + (3x + 21) \][/tex]

4. Factor by Grouping:
Now factor out the common terms from each group:

[tex]\[ = x^2(x + 7) + 3(x + 7) \][/tex]

5. Factor Common Binomial:
Notice that [tex]\( x + 7 \)[/tex] is a common binomial factor:

[tex]\[ = (x^2 + 3)(x + 7) \][/tex]

6. Combine the Factored Parts:
Finally, reintroduce the factor of 2 that we factored out initially.

[tex]\[ 2 (x^2 + 3)(x + 7) \][/tex]

Thus, the completely factored form of the polynomial [tex]\( 2x^3 + 14x^2 + 6x + 42 \)[/tex] is:

[tex]\[ 2 (x + 7)(x^2 + 3) \][/tex]