To factor the polynomial [tex]\( 8p^3q + 27q^4 \)[/tex], let's follow the steps systematically:
### Step 1: Identify a common factor
We first notice that the term [tex]\( q \)[/tex] is common in both [tex]\( 8p^3q \)[/tex] and [tex]\( 27q^4 \)[/tex]. Thus, [tex]\( q \)[/tex] can be factored out:
[tex]\[ 8p^3q + 27q^4 = q (8p^3 + 27q^3) \][/tex]
### Step 2: Recognize the remaining polynomial structure
The expression [tex]\( 8p^3 + 27q^3 \)[/tex] can be recognized as a sum of cubes since it has the form [tex]\( a^3 + b^3 \)[/tex], where:
[tex]\[ a = 2p \][/tex]
[tex]\[ b = 3q \][/tex]
### Step 3: Use the sum of cubes formula
The sum of cubes can be factored using the identity:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
Substituting [tex]\( a = 2p \)[/tex] and [tex]\( b = 3q \)[/tex] into the formula, we get:
[tex]\[ (2p)^3 + (3q)^3 = (2p + 3q)((2p)^2 - (2p)(3q) + (3q)^2) \][/tex]
### Step 4: Simplify the factored form
Now, calculate each term in the factorization:
- [tex]\( (2p + 3q) \)[/tex]
- [tex]\( (2p)^2 = 4p^2 \)[/tex]
- [tex]\( (2p)(3q) = 6pq \)[/tex]
- [tex]\( (3q)^2 = 9q^2 \)[/tex]
Thus,
[tex]\[ (2p + 3q)(4p^2 - 6pq + 9q^2) \][/tex]
### Step 5: Combine all parts
Combining the factors, including the [tex]\( q \)[/tex] we initially factored out, the fully factored form of the polynomial is:
[tex]\[ q (2p + 3q)(4p^2 - 6pq + 9q^2) \][/tex]
Therefore, the factored form of the polynomial [tex]\( 8p^3q + 27q^4 \)[/tex] is:
[tex]\[
q (2p + 3q)(4p^2 - 6pq + 9q^2)
\][/tex]
So the final answer is:
[tex]\[
\boxed{q (2p + 3q)(4p^2 - 6pq + 9q^2)}
\][/tex]
