Question

factor completely 5c^5 + 60c^4 + 180c^3. 5c^3(c + 6)(c - 6) 5c^3(c^2 + 12c + 36) 5c^3(c + 6)^2 5(c + 6)^2

Explanation:

Step1: Factor out the GCF

First, find the greatest - common factor of the terms $5c^{5}$, $60c^{4}$, and $180c^{3}$. The GCF of 5, 60, and 180 is 5, and the GCF of $c^{5}$, $c^{4}$, and $c^{3}$ is $c^{3}$. So, $5c^{5}+60c^{4}+180c^{3}=5c^{3}(c^{2}+12c + 36)$.

Step2: Factor the quadratic expression

The quadratic expression $c^{2}+12c + 36$ is a perfect - square trinomial. Since $a = 1$, $b=12$, and $c = 36$, and $b^{2}-4ac=12^{2}-4\times1\times36=144 - 144 = 0$. Also, $c^{2}+12c + 36=(c + 6)^{2}$ (because $(m + n)^{2}=m^{2}+2mn + n^{2}$, here $m = c$ and $n = 6$, $2mn=2\times c\times6 = 12c$). So, $5c^{3}(c^{2}+12c + 36)=5c^{3}(c + 6)^{2}$.

Answer:

$5c^{3}(c + 6)^{2}$