Question

Factor the polynomial

3x^4 + 6x^3 + 9x^2

Answer 1

To factor the polynomial \(3x^4 + 6x^3 + 9x^2\), we can start by finding the greatest common factor (GCF) of the coefficients and the variable terms.

1. **Identify the GCF of the coefficients**:
- The coefficients are 3, 6, and 9. The GCF of these numbers is 3.

2. **Identify the GCF of the variable terms**:
- The lowest power of \(x\) present in all terms is \(x^2\).

Combining these, the GCF of the entire polynomial is \(3x^2\).

3. **Factor out the GCF**:
\[
3x^4 + 6x^3 + 9x^2 = 3x^2 (x^2 + 2x + 3)
\]

Now we need to see if the quadratic \(x^2 + 2x + 3\) can be factored further. To see if it can be factored, we can use the discriminant method.

The discriminant \(D\) of a quadratic \(ax^2 + bx + c\) is given by:
\[
D = b^2 - 4ac
\]
Here, \(a = 1\), \(b = 2\), and \(c = 3\).

Calculating the discriminant:
\[
D = 2^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8
\]

Since the discriminant is negative (\(-8\)), the quadratic does not factor over the real numbers.

Thus, the final factorization of the polynomial \(3x^4 + 6x^3 + 9x^2\) is:
\[
\boxed{3x^2 (x^2 + 2x + 3)}
\]