Factoring the Expression 6x²+3x: Finding the Common Factor

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

Look at both the numbers and the variables separately! For $ 6x^2 + 3x $, the GCF of numbers 6 and 3 is 3, and the GCF of $ x^2 $ and $ x $ is $ x $. So the complete GCF is $ 3x $.

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

Every term must contain the GCF! If you think a term doesn't have it, look closer. In $ 6x^2 + 3x $, both terms have $ 3x $: the first term is $ 3x \cdot 2x $ and the second is $ 3x \cdot 1 $.

Q: How do I know what's left inside the parentheses?

Divide each original term by the GCF! For $ 6x^2 ÷ 3x = 2x $ and $ 3x ÷ 3x = 1 $. So you get $ 3x(2x + 1) $.

Q: Why can't I factor out just x to get x(6x + 3)?

You can factor out just x, but that's not the completely factored form! Since 6 and 3 also share a common factor of 3, you should factor out the complete GCF of $ 3x $ to get the simplest answer.

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

Use the distributive property to expand your factored form! Multiply $ 3x(2x + 1) = 3x \cdot 2x + 3x \cdot 1 = 6x^2 + 3x $. If you get back to the original expression, you're right!

Factoring Polynomials with Greatest Common Factor

We factored the expression

$6x^2+3x$

into its basic terms:

$3\cdot 2\cdot x\cdot x+3\cdot x$

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

$6x^2+3x$

into its basic terms:

$3\cdot 2\cdot x\cdot x+3\cdot x$

Take out the common factor from the factored expression

Step-by-step solution

For the expression $6x^2 + 3x$ , the greatest common factor is $3x$ : $\blue 3\cdot 2\cdot \orange x\cdot x+\blue3\cdot \orange x$

When you factor out $3x$ , the expression becomes:

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

This expresses the original expression in its factored form, with $3x(2x + 1)$ being the simplest form.

Final Answer

$3x(2x+1)$

Key Points to Remember

Essential concepts to master this topic

Rule: Find the greatest common factor (GCF) of all terms first
Technique: For $6x^2 + 3x$ , GCF is $3x$
Check: Expand $3x(2x + 1) = 6x^2 + 3x$ ✓

Common Mistakes

Avoid these frequent errors

Factoring out only numbers or only variables
Don't factor out just 3 to get 3(2x² + x) = wrong factorization! This misses the common x factor and doesn't give the simplest form. Always find the complete GCF including both numbers and variables.

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 terms with variables?

Look at both the numbers and the variables separately! For $6x^2 + 3x$ , the GCF of numbers 6 and 3 is 3, and the GCF of $x^2$ and $x$ is $x$ . So the complete GCF is $3x$ .

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

Every term must contain the GCF! If you think a term doesn't have it, look closer. In $6x^2 + 3x$ , both terms have $3x$ : the first term is $3x \cdot 2x$ and the second is $3x \cdot 1$ .

How do I know what's left inside the parentheses?

Divide each original term by the GCF! For $6x^2 ÷ 3x = 2x$ and $3x ÷ 3x = 1$ . So you get $3x(2x + 1)$ .

Why can't I factor out just x to get x(6x + 3)?

You can factor out just x, but that's not the completely factored form! Since 6 and 3 also share a common factor of 3, you should factor out the complete GCF of $3x$ to get the simplest answer.

How do I check if my factoring is correct?

Use the distributive property to expand your factored form! Multiply $3x(2x + 1) = 3x \cdot 2x + 3x \cdot 1 = 6x^2 + 3x$ . If you get back to the original expression, you're right!

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