Question

factor completely: $3x^3 - 36x^2 + 60x$

Explanation:

Step1: Factor out the GCF

The greatest common factor (GCF) of \(3x^3\), \(-36x^2\), and \(60x\) is \(3x\). So we factor out \(3x\) from each term:
\(3x^3 - 36x^2 + 60x = 3x(x^2 - 12x + 20)\)

Step2: Factor the quadratic

Now we need to factor the quadratic \(x^2 - 12x + 20\). We look for two numbers that multiply to \(20\) and add up to \(-12\). Those numbers are \(-10\) and \(-2\). So we can factor the quadratic as:
\(x^2 - 12x + 20 = (x - 10)(x - 2)\)

Step3: Combine the factors

Putting it all together, the completely factored form of the original expression is:
\(3x(x - 10)(x - 2)\)

Answer:

\(3x(x - 10)(x - 2)\)