Polynomial Factoring The Greatest Common Factor (GCF) - Free Printable

Let's solve the problem step by step based on the image you provided.

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Problem:

Factor the expression:
$$
4x^3 - 2x^2 + 6x
$$

We are asked to factor out the Greatest Common Factor (GCF) from this polynomial.

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Step 1: Break down each term into its prime factors

Let’s look at each term:

1. $ 4x^3 = 2 \cdot 2 \cdot x \cdot x \cdot x $
2. $ -2x^2 = -1 \cdot 2 \cdot x \cdot x $
3. $ 6x = 2 \cdot 3 \cdot x $

Now, identify the common factors in all three terms.

- The numerical coefficients: 4, 2, and 6
→ All divisible by 2
- The variable parts: $x^3$, $x^2$, $x$
→ All have at least one x

So, the GCF is:
$$
2x
$$

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Step 2: Factor out the GCF

We factor $2x$ from each term:

$$
4x^3 - 2x^2 + 6x = 2x( \ ?\ )
$$

Now divide each term by $2x$:

- $ \frac{4x^3}{2x} = 2x^2 $
- $ \frac{-2x^2}{2x} = -x $
- $ \frac{6x}{2x} = 3 $

So, the expression becomes:
$$
2x(2x^2 - x + 3)
$$

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✔ Final Answer:

$$
\boxed{2x(2x^2 - x + 3)}
$$

This is the factored form after taking out the GCF.

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Explanation of the Diagram:

The diagram shows:
- Each term broken into prime factors.
- The common factors ($2$ and $x$) are identified.
- The GCF is $2x$, which is factored out.
- The remaining expression inside the parentheses is what's left after dividing each term by $2x$.

That’s how we get:
$$
2x(\boxed{2x^2 - x + 3})
$$

Let me know if you want to factor further (though this trinomial doesn't factor nicely over integers).