Question

Factor of the greatest common factor from the term of the polynomial 6X ^ 3 - 12x squared+18x

Answer 1

To find the greatest common factor (GCF) of the polynomial \( 6x^3 - 12x^2 + 18x \), we first identify the coefficients and the variable factors in each term.

The polynomial has the following terms:
1. \( 6x^3 \)
2. \( -12x^2 \)
3. \( 18x \)

**Step 1: Find the GCF of the coefficients:**
- The coefficients are 6, -12, and 18.
- The factors of 6 are \( 1, 2, 3, 6 \).
- The factors of -12 are \( 1, 2, 3, 4, 6, 12 \).
- The factors of 18 are \( 1, 2, 3, 6, 9, 18 \).

The greatest common factor of \( 6, -12, \) and \( 18 \) is \( 6 \).

**Step 2: Find the GCF of the variable part:**
- The terms have variables in the following degrees:
- \( x^3 \)
- \( x^2 \)
- \( x^1 \)

The GCF of the variable part is \( x^1 \) (or simply \( x \)).

**Step 3: Combine the GCFs:**
The overall GCF of the polynomial \( 6x^3 - 12x^2 + 18x \) is:
\[
6x
\]

**Step 4: Factor out the GCF from the polynomial:**
Now we can factor \( 6x \) out of each term:

\[
6x^3 - 12x^2 + 18x = 6x(x^2 - 2x + 3)
\]

Thus, the final factored form of the polynomial is:
\[
\boxed{6x(x^2 - 2x + 3)}
\]