Question

Factor out the greatest common factor from the terms of the polynomial 6x3−12x2+18x




How to find Greatest Common Factor in polynomials.


The GCF of the polynomial given is:

Answer 1

To factor out the greatest common factor (GCF) from the polynomial \( 6x^3 - 12x^2 + 18x \), follow these steps:

1. **Identify the coefficients**: The coefficients of the terms are 6, -12, and 18.

- The GCF of the numbers \( 6, -12, \) and \( 18 \) can be found:
- Factors of 6: \( 1, 2, 3, 6 \)
- Factors of -12: \( -1, -2, -3, -4, -6, -12 \) (we can ignore the negatives for finding GCF)
- Factors of 18: \( 1, 2, 3, 6, 9, 18 \)
- The largest common factor is **6**.

2. **Identify the variable part**: The terms have variable \( x \):
- \( x^3 \) has \( x \),
- \( x^2 \) has \( x \),
- \( x \) has \( x \).

- The smallest power of \( x \) that appears in all terms is \( x^1 \).

3. **Combining the GCF**: The GCF of the polynomial \( 6x^3 - 12x^2 + 18x \) is then:
\[
6x
\]

4. **Factoring out the GCF**:
Now we can factor \( 6x \) out of each term:
\[
6x^3 - 12x^2 + 18x = 6x(x^2 - 2x + 3)
\]

So the factored form of the polynomial is:
\[
6x(x^2 - 2x + 3)
\]

To summarize:
- The GCF of the polynomial \( 6x^3 - 12x^2 + 18x \) is \( 6x \).