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]
