Find the Common Factor of 2ax+4x²: Step-by-Step Solution

Q: How do I find the greatest common factor of algebraic terms?

Look at both the coefficients and variables separately! For $2ax$ and $4x^2$: coefficients 2 and 4 have GCF of 2, and variables $x$ and $x^2$ have GCF of $x$. Combined GCF is $2x$.

Q: What if one term doesn't seem to have the common factor?

Every term must contain the common factor! If you think $2ax$ doesn't have $2x$, rewrite it as $2x \cdot a$. The factor is always there, just not obvious at first glance.

Q: Can I factor out a negative common factor?

Yes! If all terms are negative or you want to factor out a negative, you can factor out $-2x$ instead. Just remember to change the signs inside the parentheses accordingly.

Q: How do I check if my factoring is completely correct?

Use the distributive property to expand your factored form. If you get back to the original expression exactly, your factoring is perfect! For example: $2x(a + 2x) = 2ax + 4x^2$ ✓

Q: What if there's no common factor between the terms?

Then the expression cannot be factored using common factors. You would need different factoring methods like grouping or special patterns. But in this case, $2ax + 4x^2$ definitely has the common factor $2x$.

Polynomial Factoring with Common Factors

Find the common factor:

$2ax+4x^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Take out common factor

00:07 Break down 4 into factors 2 and 2

00:10 Break down the square into products

00:14 Mark the common factors

00:23 Extract the common factors from the parentheses

00:38 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the common factor:

$2ax+4x^2$

Step-by-step solution

To solve this problem, we'll employ the method of factoring by common factor:

Step 1: Identify the common factor in the terms $2ax$ and $4x^2$ .
Step 2: Factor out the common factor.
Step 3: Verify the factored form by expanding to ensure it equals the original expression.

Let's begin with Step 1:

Looking at the terms $2ax$ and $4x^2$ , we can see that both terms include $x$ as a common factor. They also share the coefficient '2'. Therefore, the greatest common factor (GCF) is $2x$ .

In Step 2, we'll factor $2x$ out of each of the terms:

$2ax$ becomes $2x \cdot a$ , and
$4x^2$ becomes $2x \cdot 2x$ .

So the expression can be written as:

$2ax + 4x^2 = 2x(a) + 2x(2x)$

Thus, factored completely using the common factor $2x$ :

$2ax + 4x^2 = 2x(a + 2x)$

In Step 3, let’s verify by expanding:

Expanding $2x(a + 2x)$ , we have:

$2x \cdot a + 2x \cdot 2x = 2ax + 4x^2$ ,
which confirms our factorization is correct.

Therefore, the solution to the problem is $2x(a + 2x)$ .

Final Answer

$2x(a+2x)$

Key Points to Remember

Essential concepts to master this topic

Greatest Common Factor: Find the largest factor that divides all terms completely
Technique: Factor out $2x$ from both $2ax$ and $4x^2$
Verification: Expand $2x(a + 2x) = 2ax + 4x^2$ to check your work ✓

Common Mistakes

Avoid these frequent errors

Factoring out only part of the common factor
Don't factor out just $x$ when $2x$ is the greatest common factor = $x(2a + 4x)$ instead of $2x(a + 2x)$ ! You miss the coefficient 2 that's common to both terms. Always identify the complete greatest common factor including both numerical coefficients and variable parts.

Practice Quiz

Test your knowledge with interactive questions

Break down the expression into basic terms:

$$ 2x^2 $$

FAQ

Everything you need to know about this question

How do I find the greatest common factor of algebraic terms?

Look at both the coefficients and variables separately! For $2ax$ and $4x^2$ : coefficients 2 and 4 have GCF of 2, and variables $x$ and $x^2$ have GCF of $x$ . Combined GCF is $2x$ .

What if one term doesn't seem to have the common factor?

Every term must contain the common factor! If you think $2ax$ doesn't have $2x$ , rewrite it as $2x \cdot a$ . The factor is always there, just not obvious at first glance.

Can I factor out a negative common factor?

Yes! If all terms are negative or you want to factor out a negative, you can factor out $-2x$ instead. Just remember to change the signs inside the parentheses accordingly.

How do I check if my factoring is completely correct?

Use the distributive property to expand your factored form. If you get back to the original expression exactly, your factoring is perfect! For example: $2x(a + 2x) = 2ax + 4x^2$ ✓

What if there's no common factor between the terms?

Then the expression cannot be factored using common factors. You would need different factoring methods like grouping or special patterns. But in this case, $2ax + 4x^2$ definitely has the common factor $2x$ .

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