Question

What is the best method for fully factoring this polynomial 3x^3-7x^2-6x
Gcf and factoring by grouping
Differences of squares
Gcf only
Gcf and busting b's trinomial

Answer 1

To fully factor the polynomial \( 3x^3 - 7x^2 - 6x \), we can start by identifying the greatest common factor (GCF).

1. **Identify the GCF**:
The GCF of the terms \( 3x^3 \), \( -7x^2 \), and \( -6x \) is \( x \).

2. **Factor out the GCF**:
\[
3x^3 - 7x^2 - 6x = x(3x^2 - 7x - 6)
\]

3. **Now factor the quadratic \( 3x^2 - 7x - 6 \)**:
To factor this quadratic, we look for two numbers that multiply to \( 3 \times -6 = -18 \) and add to \( -7 \). The numbers that satisfy this are \( -9 \) and \( 2 \).

Rewrite the middle term:
\[
3x^2 - 9x + 2x - 6
\]

4. **Group the terms**:
\[
(3x^2 - 9x) + (2x - 6)
\]

Factor each group:
\[
3x(x - 3) + 2(x - 3)
\]

5. **Factor out the common binomial**:
\[
(3x + 2)(x - 3)
\]

6. **Combine everything together**:
Therefore, the fully factored form of \( 3x^3 - 7x^2 - 6x \) is:
\[
x(3x + 2)(x - 3)
\]

**Final Answer**:
\[
x(3x + 2)(x - 3)
\]

The best method used here involves finding the GCF and factoring by grouping. So, the correct approach is **GCF and factoring by grouping**.