Finding Common Factors in 5x²+10x: Step-by-Step Factoring

Q: How do I know if 5x is really the greatest common factor?

Check each term: $ 5x^2 = 5 \cdot x \cdot x $ contains both 5 and x, while $ 10x = 2 \cdot 5 \cdot x $ also contains both. Since both terms have 5 and x, the GCF is $ 5x $.

Q: What's wrong with just factoring out 5?

Factoring out only 5 gives you $ 5(x^2 + 2x) $, but you can factor even more! The expression $ x^2 + 2x $ still has a common factor of x. Always find the greatest common factor to fully simplify.

Q: How do I check my factoring is correct?

Multiply your factored form back out! If $ 5x(x + 2) $ is correct, then $ 5x \cdot x + 5x \cdot 2 = 5x^2 + 10x $. It matches the original!

Q: Why do we break terms into basic factors first?

Breaking $ 5x^2 $ into $ 5 \cdot x \cdot x $ and $ 10x $ into $ 2 \cdot 5 \cdot x $ helps you see all the factors clearly. This makes it easier to identify what's common between terms.

Polynomial Factoring with Greatest Common Factor

We factored the expression

$5x^2+10x$ into its basic terms:

$5\cdot x\cdot x+2\cdot5\cdot x$

What common factor can be found in these terms?

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

We factored the expression

$5x^2+10x$ into its basic terms:

$5\cdot x\cdot x+2\cdot5\cdot x$

What common factor can be found in these terms?

Step-by-step solution

First, consider the expression $5x^2+10x$ . We want to factor out the greatest common factor of the terms.

Write each term showing the factor $x$ : $\blue 5\cdot \orange x\cdot x$ and $2\cdot\blue5\cdot \orange x$ .

Both terms, $5x^2$ and $10x$ , contain the factor $x$ as well as $5$ . Therefore, $5\cdot x$ is a common factor.

The greatest common factor is $5\cdot x$ .

Final Answer

$5\cdot x$

Key Points to Remember

Essential concepts to master this topic

Greatest Common Factor: Find the largest factor shared by all terms
Technique: Break down $5x^2$ as $5 \cdot x \cdot x$ and $10x$ as $2 \cdot 5 \cdot x$
Verify: Factor out $5x$ gives $5x(x + 2)$ , expand back to check ✓

Common Mistakes

Avoid these frequent errors

Taking only the numerical coefficient as the common factor
Don't just factor out 5 and ignore the variable x! This gives 5(x² + 2x) which still isn't fully factored. You miss the complete greatest common factor. Always look for both numerical AND variable factors that appear in every term.

Practice Quiz

Test your knowledge with interactive questions

Break down the expression into basic terms:

$$ 4x^2 + 6x $$

FAQ

Everything you need to know about this question

How do I know if 5x is really the greatest common factor?

Check each term: $5x^2 = 5 \cdot x \cdot x$ contains both 5 and x, while $10x = 2 \cdot 5 \cdot x$ also contains both. Since both terms have 5 and x, the GCF is $5x$ .

What's wrong with just factoring out 5?

Factoring out only 5 gives you $5(x^2 + 2x)$ , but you can factor even more! The expression $x^2 + 2x$ still has a common factor of x. Always find the greatest common factor to fully simplify.

How do I check my factoring is correct?

Multiply your factored form back out! If $5x(x + 2)$ is correct, then $5x \cdot x + 5x \cdot 2 = 5x^2 + 10x$ . It matches the original!

Why do we break terms into basic factors first?

Breaking $5x^2$ into $5 \cdot x \cdot x$ and $10x$ into $2 \cdot 5 \cdot x$ helps you see all the factors clearly. This makes it easier to identify what's common between terms.

Can there be more than one common factor?

Yes! In this problem, both 5 and x are common factors, but we want the greatest common factor, which combines all common factors: $5x$ .

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