Find the GCF of 12x+16y: Step-by-Step Factor Analysis

Q: How do I know if 4 is really the biggest common factor?

Check by prime factorization: 12 = 2²×3 and 16 = 2⁴. The GCF uses the lowest power of each common prime, so GCF = 2² = 4.

Q: What if the variables had common factors too?

In $ 12x+16y $, the variables x and y are different, so they don't share common factors. Only look at the coefficients 12 and 16.

Q: How do I check if my factored answer is correct?

Expand your answer using the distributive property: 4(3x+4y) = 4×3x + 4×4y = 12x + 16y ✓

Factoring Expressions with Coefficient Analysis

Find the biggest common factor:

$12x+16y$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find a common factor

00:07 Factor 12 into factors 4 and 3

00:11 Factor 12 into factors 4 and 4

00:14 Mark the common factors

00:21 Take out the common factors from the parentheses

00:37 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 biggest common factor:

$12x+16y$

Step-by-step solution

We begin by breaking down the coefficients 12 and 16 into multiplication exercises with a multiplying factor to eventually simplify:

$3\times4\times x+4\times4\times y$

We then extract 4 which is the common factor:

$4(3\times x+4\times y)=4(3x+4y)$

Final Answer

$4(3x+4y)$

Key Points to Remember

Essential concepts to master this topic

Prime Factorization: Break down coefficients to find all possible common factors
GCF Method: Factor 12 = 4×3 and 16 = 4×4, so GCF is 4
Verification: Check by expanding: 4(3x+4y) = 12x+16y ✓

Common Mistakes

Avoid these frequent errors

Taking partial common factors instead of the greatest
Don't factor out just 2 to get 2(6x+8y) = still has common factor 2! This leaves additional factoring undone and isn't fully simplified. Always find the GCF by comparing all prime factors completely.

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 4 is really the biggest common factor?

Check by prime factorization: 12 = 2²×3 and 16 = 2⁴. The GCF uses the lowest power of each common prime, so GCF = 2² = 4.

What if the variables had common factors too?

In $12x+16y$ , the variables x and y are different, so they don't share common factors. Only look at the coefficients 12 and 16.

Why can't I factor out 12 to get 12x(1 + 4y/3)?

You can't factor 12 from 16y because 16 ÷ 12 = 4/3, which creates a fraction inside the parentheses. The GCF must divide evenly into both terms.

How do I check if my factored answer is correct?

Expand your answer using the distributive property: 4(3x+4y) = 4×3x + 4×4y = 12x + 16y ✓

What if I can't see the common factor right away?

List the factors of each coefficient: 12 = {1,2,3,4,6,12} and 16 = {1,2,4,8,16}. The largest number in both lists is your GCF!

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