To factor out the greatest common factor (GCF) from the expression [tex]\( x^3 - 2x^2 - 3x \)[/tex], follow these steps:
1. Identify the GCF of the terms:
- Each term in the expression [tex]\( x^3, -2x^2, \)[/tex] and [tex]\( -3x \)[/tex] has a common factor which is [tex]\( x \)[/tex].
2. Factor out the GCF [tex]\( x \)[/tex]:
- When you factor [tex]\( x \)[/tex] out of each term, you get:
[tex]\[
x^3 - 2x^2 - 3x = x(x^2 - 2x - 3)
\][/tex]
3. Factor the quadratic expression inside the parentheses:
- The quadratic expression is [tex]\( x^2 - 2x - 3 \)[/tex].
- To factor [tex]\( x^2 - 2x - 3 \)[/tex], look for two numbers that multiply to [tex]\(-3\)[/tex] (the constant term) and add to [tex]\(-2\)[/tex] (the coefficient of the [tex]\( x \)[/tex] term).
4. Determine the factors:
- The numbers [tex]\(-3\)[/tex] and [tex]\(1\)[/tex] multiply to [tex]\(-3\)[/tex] and add up to [tex]\(-2\)[/tex].
- Therefore, [tex]\( x^2 - 2x - 3 \)[/tex] can be factored as [tex]\( (x - 3)(x + 1) \)[/tex].
5. Combine the factored terms:
- The original expression [tex]\( x^3 - 2x^2 - 3x \)[/tex] can now be written as:
[tex]\[
x^3 - 2x^2 - 3x = x(x - 3)(x + 1)
\][/tex]
So, the factorized form of the expression [tex]\( x^3 - 2x^2 - 3x \)[/tex] is:
[tex]\[
x(x - 3)(x + 1)
\][/tex]
