Factor the Expression: 14xyz + 8x²y³z Step by Step

Q: How do I find the GCF of the coefficients 14 and 8?

List the factors: 14 = 2 × 7 and 8 = 2 × 4. The greatest common factor is 2, the largest number that divides both evenly.

Q: Why is the GCF xyz and not x²y³z?

Take the lowest power of each variable that appears in all terms. Since 14xyz has x¹, y¹, z¹ and 8x²y³z has x², y³, z¹, the GCF uses x¹y¹z¹ = xyz.

Q: What if I factored out the wrong GCF?

You can check by multiplying your factored form back out. If you don't get the original expression, try again with the complete GCF.

Q: Can I factor this expression further after finding the GCF?

After factoring out $ 2xyz $, you get $ 2xyz(7 + 4xy^2) $. The expression 7 + 4xy² has no common factors, so this is completely factored.

Q: Why do some answer choices have different coefficients outside the parentheses?

Those are incorrect factorizations! Only $ 2xyz $ is the complete GCF. Other coefficients like 4xyz or 8xyz don't divide evenly into both original terms.

Polynomial Factoring with Greatest Common Factor

Decompose the following expression into factors:

$14xyz+8x^2y^3z$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 First, let's find the common factors.

00:11 Next, factor 14 into 7 and 2. Great job.

00:19 Now, factor 8 into 4 and 2. You're doing well.

00:24 Let's break down the square into its product components.

00:28 Break down the power of 3 into a square and a product.

00:34 Identify the common factors we marked earlier. Keep it up.

00:45 Take out those common factors from the parentheses.

01:02 And that's how we solve this problem. Well done!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Decompose the following expression into factors:

$14xyz+8x^2y^3z$

Step-by-step solution

First, we break down all the powers into multiplication exercises and at the same time try to reduce the integers as much as possible:

7*2*xyz+2*4*x*x*y*y²*z

Now we use the substitution property to arrange the equation into a more manageable form:

2*x*y*z*7+2*x*y*z*x*y²

Lastly we try to find the common factor among all the parts - 2xyz

2xyz(7+xy²)

Final Answer

$2xyz(7+4xy^2)$

Key Points to Remember

Essential concepts to master this topic

GCF Rule: Find largest common factor among all coefficients and variables
Technique: Factor out GCF first: 2xyz from both 14xyz and 8x²y³z
Check: Multiply factored form back: 2xyz(7 + 4xy²) = 14xyz + 8x²y³z ✓

Common Mistakes

Avoid these frequent errors

Not finding the complete GCF before factoring
Don't factor out just part of the GCF like xyz instead of 2xyz = incomplete factorization! This leaves extra common factors inside the parentheses that should have been factored out. Always identify the complete GCF including numerical coefficients and all common variables with their lowest powers.

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 GCF of the coefficients 14 and 8?

List the factors: 14 = 2 × 7 and 8 = 2 × 4. The greatest common factor is 2, the largest number that divides both evenly.

Why is the GCF xyz and not x²y³z?

Take the lowest power of each variable that appears in all terms. Since 14xyz has x¹, y¹, z¹ and 8x²y³z has x², y³, z¹, the GCF uses x¹y¹z¹ = xyz.

What if I factored out the wrong GCF?

You can check by multiplying your factored form back out. If you don't get the original expression, try again with the complete GCF.

Can I factor this expression further after finding the GCF?

After factoring out $2xyz$ , you get $2xyz(7 + 4xy^2)$ . The expression 7 + 4xy² has no common factors, so this is completely factored.

Why do some answer choices have different coefficients outside the parentheses?

Those are incorrect factorizations! Only $2xyz$ is the complete GCF. Other coefficients like 4xyz or 8xyz don't divide evenly into both original terms.

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