$$ \frac{ ext { Factor }}{18 x^{3}+12 x^{2}+2 x} $$

Question

$$ \frac{	ext { Factor }}{18 x^{3}+12 x^{2}+2 x} $$

UpStudy AI · Accepted Answer

Sure, let's factor the expression in the denominator:

$$
18x^3 + 12x^2 + 2x
$$

**Step 1: Factor Out the Greatest Common Factor (GCF)**

First, identify the GCF of the terms $$18x^3$$, $$12x^2$$, and $$2x$$.

- The GCF of the coefficients $$18$$, $$12$$, and $$2$$ is $$2$$.
- The smallest power of $$x$$ present in all terms is $$x$$.

So, factor out $$2x$$:

$$
18x^3 + 12x^2 + 2x = 2x(9x^2 + 6x + 1)
$$

**Step 2: Factor the Quadratic Expression**

Now, let's factor the quadratic expression inside the parentheses:

$$
9x^2 + 6x + 1
$$

To factor this, look for two numbers that multiply to $$9 \times 1 = 9$$ and add up to $$6$$. The numbers $$3$$ and $$3$$ fit this criteria.

So, rewrite the quadratic:

$$
9x^2 + 6x + 1 = (3x + 1)(3x + 1) = (3x + 1)^2
$$

**Step 3: Combine the Factors**

Putting it all together:

$$
18x^3 + 12x^2 + 2x = 2x(3x + 1)^2
$$

**Final Factored Form:**

$$
\frac{\text{Factor}}{18x^{3}+12x^{2}+2x} = \frac{\text{Factor}}{2x(3x + 1)^2}
$$

If your goal was to factor the denominator, the fully factored form is:

$$
2x(3x + 1)^2
$$

Feel free to let me know if you need further assistance!
The denominator factors to $$2x(3x + 1)^2$$.

\( \frac{\text { Factor }}{18 x^{3}+12 x^{2}+2 x} \)