Find the Common Factor in 5x² + 25x: Polynomial Factoring Practice

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

Look at both coefficients and variables separately. For $ 5x^2 + 25x $: coefficients 5 and 25 share factor 5, variables x² and x share factor x. So GCF = 5x.

Q: Why can't I just factor out x from both terms?

You can factor out x, but it's not the greatest common factor! You'd get $ x(5x + 25) $, but notice 5x and 25 still share factor 5. Always find the largest possible factor.

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

Use the distributive property to expand your answer. If $ 5x(x + 5) = 5x \cdot x + 5x \cdot 5 = 5x^2 + 25x $ matches the original, you're right!

Q: What does 'take out the common factor' actually mean?

It means divide each term by the GCF and write it as multiplication. Like taking $ 5x $ out of $ 5x^2 $ leaves $ x $, and out of $ 25x $ leaves $ 5 $.

Polynomial Factoring with Greatest Common Factor

We factored the expression $5x^2 + 25x$ into its basic terms:

$5 \cdot x \cdot x + 25 \cdot x$

Take out the common factor from the factored expression

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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 $5x^2 + 25x$ into its basic terms:

$5 \cdot x \cdot x + 25 \cdot x$

Take out the common factor from the factored expression

Step-by-step solution

To factor the expression $5x^2 + 25x$ , we look for the greatest common factor (GCF) of the terms $5x^2$ and $25x$ . The GCF is $5x$ . We factor out $5x$ from each term:

$5x^2+25x=\blue5\cdot \orange x\cdot x+\blue 5\cdot5\cdot \orange x$ .

This results in the expression $5x(x + 5)$ .

Final Answer

$5x(x + 5)$

Key Points to Remember

Essential concepts to master this topic

GCF Rule: Find the largest factor shared by all terms
Technique: Factor out $5x$ from $5x^2 + 25x$ gives $5x(x + 5)$
Check: Expand $5x(x + 5) = 5x^2 + 25x$ matches original ✓

Common Mistakes

Avoid these frequent errors

Only factoring out the coefficient and ignoring variables
Don't factor out just 5 and get 5(x² + 5x) = wrong factorization! This misses the common x factor in both terms. Always find the complete GCF including all shared variables: 5x from both 5x² and 25x.

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 find the greatest common factor of polynomial terms?

Look at both coefficients and variables separately. For $5x^2 + 25x$ : coefficients 5 and 25 share factor 5, variables x² and x share factor x. So GCF = 5x.

What if I factor out the wrong amount?

Your remaining expression in parentheses will tell you! If you see common factors left inside like $5(x^2 + 5x)$ , you didn't factor enough. Keep going until no common factors remain.

Why can't I just factor out x from both terms?

You can factor out x, but it's not the greatest common factor! You'd get $x(5x + 25)$ , but notice 5x and 25 still share factor 5. Always find the largest possible factor.

How do I check if my factoring is correct?

Use the distributive property to expand your answer. If $5x(x + 5) = 5x \cdot x + 5x \cdot 5 = 5x^2 + 25x$ matches the original, you're right!

What does 'take out the common factor' actually mean?

It means divide each term by the GCF and write it as multiplication. Like taking $5x$ out of $5x^2$ leaves $x$ , and out of $25x$ leaves $5$ .

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