Factor the Expression 3y + 6y: Finding the Greatest Common Factor

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

Look at both the numbers and the variables separately! For $ 3y + 6y $, the GCF of 3 and 6 is 3, and both terms have y, so the GCF is 3y.

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

You can, but it's not the greatest common factor! Factoring out just y gives $ y(3 + 6) $, but 3 and 6 still share a common factor of 3. Always factor out the complete GCF.

Q: What do I put in the parentheses after factoring?

Whatever is left over after dividing each term by the GCF! For $ 3y ÷ 3y = 1 $ and $ 6y ÷ 3y = 2 $, so you get $ 3y(1 + 2) $.

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

Use the distributive property to multiply back out! If $ 3y(1 + 2) = 3y \cdot 1 + 3y \cdot 2 = 3y + 6y $, then your factoring is correct.

Q: What if the terms don't have any common factors?

If there's truly no common factor, then the expression is already in simplest form! But always double-check by looking at coefficients and variables separately.

Factoring Expressions with Common Factors

We factored the expression $3y + 6y$ into its basic terms:

$3\cdot y + 6\cdot y$

Take out the greatest 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 $3y + 6y$ into its basic terms:

$3\cdot y + 6\cdot y$

Take out the greatest common factor from the factored expression.

Step-by-step solution

We start with the expression $3y + 6y$ .

First, we notice that both terms share a common factor of $y$ .

So, we factor out $y$ from each term:

$3y = y\cdot 3$ and $6y = y\cdot 6$ .

We can also see that $6$ can be factored to $2\cdot3$ .
Now when we look at the expression we can see that both $y$ and $3$ are the common factor: $\blue3\cdot \orange y+2\cdot\blue 3\cdot \orange y$

This allows us to rewrite the expression as $3y\left(1+2\right)$ , as nothing is left from the first term, and so we keep there a $1$ , and $2$ is left from the second term.

Thus, the factored form is $3y\left(1+2\right)$

Final Answer

$3y\left(1+2\right)$

Key Points to Remember

Essential concepts to master this topic

Greatest Common Factor: Find the largest factor shared by all terms
Technique: $3y + 6y = 3y(1 + 2)$ by factoring out 3y
Check: Distribute back: $3y(1 + 2) = 3y + 6y$ ✓

Common Mistakes

Avoid these frequent errors

Factoring out only part of the common factor
Don't factor out just y when 3y is the greatest common factor = $y(3 + 6)$ instead of $3y(1 + 2)$ ! This leaves additional common factors unfactored. Always find the complete greatest common factor by checking both numerical and variable parts.

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 algebraic terms?

Look at both the numbers and the variables separately! For $3y + 6y$ , the GCF of 3 and 6 is 3, and both terms have y, so the GCF is 3y.

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

You can, but it's not the greatest common factor! Factoring out just y gives $y(3 + 6)$ , but 3 and 6 still share a common factor of 3. Always factor out the complete GCF.

What do I put in the parentheses after factoring?

Whatever is left over after dividing each term by the GCF! For $3y ÷ 3y = 1$ and $6y ÷ 3y = 2$ , so you get $3y(1 + 2)$ .

How can I check if my factoring is correct?

Use the distributive property to multiply back out! If $3y(1 + 2) = 3y \cdot 1 + 3y \cdot 2 = 3y + 6y$ , then your factoring is correct.

What if the terms don't have any common factors?

If there's truly no common factor, then the expression is already in simplest form! But always double-check by looking at coefficients and variables separately.

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