Decompose the Expression: 15xyz² + 25xy/z Step by Step

Q: How do I find the GCF when one term has z² and the other has 1/z?

Great question! Since one term has $ z^2 $ and the other has $ \frac{1}{z} $, there's no common z factor. Only factor out what appears in both terms: the coefficient 5 and variables x and y.

Q: Why is the answer 5xy and not 15xy?

The GCF of 15 and 25 is 5, not 15! Think of it this way: 15 = 3 × 5 and 25 = 5 × 5. The largest number that divides both is 5.

Q: How do I handle the fraction xy/z when factoring?

Treat $ \frac{xy}{z} $ as $ xy \cdot \frac{1}{z} $. You can factor out the xy part, leaving $ \frac{1}{z} $ inside the parentheses. So $ 25\frac{xy}{z} = 5xy \cdot \frac{5}{z} $.

Q: Can I check my answer by expanding it back?

Yes, always do this! Distribute $ 5xy $ through the parentheses: $ 5xy \cdot 3z^2 = 15xyz^2 $ and $ 5xy \cdot \frac{5}{z} = 25\frac{xy}{z} $. You should get back the original expression.

Q: What if I factored out something different like xy?

You'd get a partially factored form like $ xy(15z^2 + \frac{25}{z}) $. This isn't wrong, but it's not fully factored since you can still factor out 5 from the parentheses to get the complete answer.

Factoring Expressions with Mixed Variable Powers

Which of the expressions is a decomposition of the simplified expression below?

$15xyz^2+25\frac{xy}{z}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the common factor

00:03 Factor 15 into factors 5 and 3

00:10 Factor 25 into factors 5 and 5

00:17 Mark the common factors

00:27 Take out the common factors from the parentheses

00:48 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

Which of the expressions is a decomposition of the simplified expression below?

$15xyz^2+25\frac{xy}{z}$

Step-by-step solution

To solve the problem, we aim to factor the expression by identifying the greatest common factor (GCF) for the terms $15xyz^2$ and $25\frac{xy}{z}$ .

Step 1: Identify the GCF
- Both terms contain the factors $x$ and $y$ .
- The first term $15xyz^2$ consists of $15$ , $x$ , $y$ , and $z^2$ .
- The second term $25\frac{xy}{z}$ consists of $25$ , $x$ , $y$ , and $\frac{1}{z}$ .
- The GCF of the constants 15 and 25 is 5.

Step 2: Factor out the GCF
- The GCF of the variables is $xy$ .
- Therefore, the overall GCF we can factor out is $5xy$ .

Step 3: Simplify the remaining expression
- Factoring out $5xy$ from the expression:
$15xyz^2 = 5xy \times 3z^2$
$25\frac{xy}{z} = 5xy \times \frac{5}{z}$

Step 4: Write the factored expression
This gives us:
$15xyz^2 + 25\frac{xy}{z} = 5xy(3z^2 + \frac{5}{z})$

Thus, the decomposition of the simplified expression is $5xy(3z^2 + \frac{5}{z})$ , which corresponds to choice 2.

Therefore, the solution to the problem is $5xy(3z^2 + \frac{5}{z})$ .

Final Answer

$5xy(3z^2+\frac{5}{z})$

Key Points to Remember

Essential concepts to master this topic

Greatest Common Factor: Find GCF of coefficients and common variables first
Technique: Factor out $5xy$ from both $15xyz^2$ and $25\frac{xy}{z}$
Check: Expand factored form: $5xy(3z^2 + \frac{5}{z}) = 15xyz^2 + 25\frac{xy}{z}$ ✓

Common Mistakes

Avoid these frequent errors

Factoring out coefficients without considering all variables
Don't factor out just 5 and ignore the xy variables = incomplete factorization! This leaves extra variables in the parentheses that could be factored out. Always identify ALL common factors including coefficients AND variables before factoring.

Practice Quiz

Test your knowledge with interactive questions

Break down the expression into basic terms:

$$ 2x^2 $$

FAQ

Everything you need to know about this question

How do I find the GCF when one term has z² and the other has 1/z?

Great question! Since one term has $z^2$ and the other has $\frac{1}{z}$ , there's no common z factor. Only factor out what appears in both terms: the coefficient 5 and variables x and y.

Why is the answer 5xy and not 15xy?

The GCF of 15 and 25 is 5, not 15! Think of it this way: 15 = 3 × 5 and 25 = 5 × 5. The largest number that divides both is 5.

How do I handle the fraction xy/z when factoring?

Treat $\frac{xy}{z}$ as $xy \cdot \frac{1}{z}$ . You can factor out the xy part, leaving $\frac{1}{z}$ inside the parentheses. So $25\frac{xy}{z} = 5xy \cdot \frac{5}{z}$ .

Can I check my answer by expanding it back?

Yes, always do this! Distribute $5xy$ through the parentheses: $5xy \cdot 3z^2 = 15xyz^2$ and $5xy \cdot \frac{5}{z} = 25\frac{xy}{z}$ . You should get back the original expression.

What if I factored out something different like xy?

You'd get a partially factored form like $xy(15z^2 + \frac{25}{z})$ . This isn't wrong, but it's not fully factored since you can still factor out 5 from the parentheses to get the complete answer.

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