Question

factor $3m^{4}+81mv^{3}$ completely.

Explanation:

Step1: Factor out GCF

Identify and factor out the greatest common factor (GCF) of the two terms. The GCF of $3m^4$ and $81mv^3$ is $3m$.
$3m^4 + 81mv^3 = 3m(m^3 + 27v^3)$

Step2: Recognize sum of cubes

Notice that $m^3 + 27v^3$ is a sum of perfect cubes, where $m^3 = m^3$ and $27v^3 = (3v)^3$. Use the sum of cubes formula: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.
$m^3 + 27v^3 = m^3 + (3v)^3 = (m + 3v)(m^2 - 3mv + 9v^2)$

Step3: Combine all factors

Substitute the factored form of the sum of cubes back into the expression from Step1.
$3m(m^3 + 27v^3) = 3m(m + 3v)(m^2 - 3mv + 9v^2)$

Answer:

$3m(m + 3v)(m^2 - 3mv + 9v^2)$