Question

factor $2t^{5}-24t^{4}+72t^{3}$ completely.

Explanation:

Step1: Factor out the GCF

First, find the greatest common factor (GCF) of the terms \(2t^5\), \(-24t^4\), and \(72t^3\). The GCF of the coefficients \(2\), \(-24\), and \(72\) is \(2\), and the GCF of the variable parts \(t^5\), \(t^4\), and \(t^3\) is \(t^3\). So we factor out \(2t^3\):
\(2t^5 - 24t^4 + 72t^3 = 2t^3(t^2 - 12t + 36)\)

Step2: Factor the quadratic

Now, factor the quadratic \(t^2 - 12t + 36\). We need two numbers that multiply to \(36\) and add up to \(-12\). Those numbers are \(-6\) and \(-6\), so:
\(t^2 - 12t + 36 = (t - 6)(t - 6) = (t - 6)^2\)

Step3: Combine the factors

Putting it all together, the completely factored form is:
\(2t^3(t - 6)^2\)

Answer:

\(2t^3(t - 6)^2\)