Factor the Expression: 256xy³-32zy² Step-by-Step

Q: How do I find the GCF when there are different variables like x and z?

Only factor out variables that appear in every term. Since x appears only in the first term and z only in the second, we can't factor them out. But $ y^2 $ appears in both terms, so it goes in the GCF.

Q: Why is the GCF 32y² and not 8y²?

We need the greatest common factor! Since 256 = 32 × 8 and 32 = 32 × 1, the largest number that divides both is 32. For variables, $ y^3 $ and $ y^2 $ share $ y^2 $ as their greatest common power.

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

Use the distributive property to expand your answer! Multiply $ 32y^2 $ by each term inside the parentheses: $ 32y^2 \times 8xy = 256xy^3 $ and $ 32y^2 \times (-z) = -32zy^2 $.

Q: Can I factor this expression further?

Not really! The expression inside the parentheses $ (8xy - z) $ has no common factors between 8xy and z, so $ 32y^2(8xy - z) $ is fully factored.

Factoring Expressions with Multiple Variables

Factor the following expression:

$256xy^3-32zy^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the common factor

00:04 Factorize 256 into factors 32 and 8

00:09 Break down power of 3 into square and product

00:16 Mark the common factors

00:25 Take out the common factors from parentheses

00:38 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

Factor the following expression:

$256xy^3-32zy^2$

Step-by-step solution

To solve this problem, we'll follow these steps to factor $256xy^3 - 32zy^2$ :

Step 1: Identify the greatest common factor (GCF) for both terms.
Step 2: Factor out the GCF from the expression.

Now, let's work through each step:

Step 1: Identify the GCF.
The first term is $256xy^3$ , and the second term is $32zy^2$ .

The numerical GCF of 256 and 32 is 32.
The terms $xy^3$ and $zy^2$ have a common factor of $y^2$ .

Thus, the GCF of the entire expression is $32y^2$ .

Step 2: Factor out the GCF.
We'll factor $32y^2$ out of each term:

The first term $256xy^3$ becomes $32y^2 \times 8xy$ . (since $\frac{256xy^3}{32y^2} = 8xy$ )
The second term $32zy^2$ becomes $32y^2 \times z$ . (since $\frac{32zy^2}{32y^2} = z$ )

Therefore, factoring the expression by the GCF gives us:
$256xy^3 - 32zy^2 = 32y^2(8xy - z)$ .

The factorized form of the expression is $32y^2(8xy - z)$ .

Final Answer

$32y^2(8xy-z)$

Key Points to Remember

Essential concepts to master this topic

GCF Rule: Find greatest common factor of all coefficients and variables
Technique: Factor out $32y^2$ from $256xy^3$ gives $8xy$
Check: Expand $32y^2(8xy - z) = 256xy^3 - 32zy^2$ ✓

Common Mistakes

Avoid these frequent errors

Factoring out only the numerical GCF
Don't factor out just 32 and ignore the variables = incomplete factoring! This leaves unfactored variable terms that could be simplified further. Always find the GCF of both numbers AND variables together.

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 there are different variables like x and z?

Only factor out variables that appear in every term. Since x appears only in the first term and z only in the second, we can't factor them out. But $y^2$ appears in both terms, so it goes in the GCF.

Why is the GCF 32y² and not 8y²?

We need the greatest common factor! Since 256 = 32 × 8 and 32 = 32 × 1, the largest number that divides both is 32. For variables, $y^3$ and $y^2$ share $y^2$ as their greatest common power.

What if I can't see the GCF right away?

Break it down step by step! First find the GCF of the numbers (256 and 32), then find the GCF of each variable separately. For y-terms: $y^3$ and $y^2$ share $y^2$ .

How do I check if my factoring is correct?

Use the distributive property to expand your answer! Multiply $32y^2$ by each term inside the parentheses: $32y^2 \times 8xy = 256xy^3$ and $32y^2 \times (-z) = -32zy^2$ .

Can I factor this expression further?

Not really! The expression inside the parentheses $(8xy - z)$ has no common factors between 8xy and z, so $32y^2(8xy - z)$ is fully factored.

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