Answer :

To factor the given cubic expression [tex]\(3x^3 - 48x^2 + 180x\)[/tex] completely, follow these steps:

1. Identify the Greatest Common Factor (GCF):

First, look for the GCF of all the terms in the expression. The terms are [tex]\(3x^3\)[/tex], [tex]\(-48x^2\)[/tex], and [tex]\(180x\)[/tex].

The GCF is [tex]\(3x\)[/tex]. This is because:

- Each term has at least one factor of [tex]\(x\)[/tex].
- The coefficients 3, 48, and 180 have a common factor of 3.

2. Factor Out the GCF:

Factor [tex]\(3x\)[/tex] out of each term in the expression:

[tex]\[
3x^3 - 48x^2 + 180x = 3x(x^2 - 16x + 60)
\][/tex]

3. Factor the Quadratic Expression:

Now, focus on the quadratic expression inside the parentheses: [tex]\(x^2 - 16x + 60\)[/tex].

We need to find two numbers that multiply to 60 and add up to -16.

The numbers that satisfy this are -10 and -6 because:

- [tex]\(-10 \times -6 = 60\)[/tex]
- [tex]\(-10 + (-6) = -16\)[/tex]

Using these numbers, factor the quadratic:

[tex]\[
x^2 - 16x + 60 = (x - 10)(x - 6)
\][/tex]

4. Write the Fully Factored Form:

Substitute the factored quadratic back into the expression:

[tex]\[
3x(x^2 - 16x + 60) = 3x(x - 10)(x - 6)
\][/tex]

Therefore, the completely factored form of the expression [tex]\(3x^3 - 48x^2 + 180x\)[/tex] is:

[tex]\[
3x(x - 10)(x - 6)
\][/tex]

This is the final answer.