Let's factor the polynomial [tex]\(2x^3 + 7x^2 + 2x + 14\)[/tex] completely step-by-step.
### Step 1: Identify the polynomial
The polynomial we need to factor is:
[tex]\[ 2x^3 + 7x^2 + 2x + 14 \][/tex]
### Step 2: Factor out the greatest common factor (GCF)
First, we look for the greatest common factor (GCF) among the terms. However, in this case, there is no common factor for all terms.
### Step 3: Group terms to factor by grouping
We try to group terms in a way that makes it easier to factor by grouping:
[tex]\[ 2x^3 + 7x^2 + 2x + 14 \][/tex]
Group the terms:
[tex]\[ (2x^3 + 7x^2) + (2x + 14) \][/tex]
### Step 4: Factor out the GCF in each group
Factor out the GCF from each group:
[tex]\[ x^2 (2x + 7) + 2(2x + 7) \][/tex]
### Step 5: Factor common binomial factor
Now we can see that [tex]\((2x + 7)\)[/tex] is a common factor:
[tex]\[ (2x + 7)(x^2 + 2) \][/tex]
### Step 6: Check for further factorization
Check if [tex]\(x^2 + 2\)[/tex] can be factored further. Since [tex]\(x^2 + 2\)[/tex] does not factor further over the real numbers, we have our complete factorization.
### Final Solution
Thus, the completely factored form of the polynomial [tex]\(2x^3 + 7x^2 + 2x + 14\)[/tex] is:
[tex]\[ (2x + 7)(x^2 + 2) \][/tex]
This is the fully factored form of the given polynomial.
