Decompose the Expression: xy/2 + x/4y into Factors

Q: How do I find the common factor when terms have different denominators?

Look at each term separately: $ \frac{xy}{2} = \frac{x}{2} \cdot y $ and $ \frac{x}{4y} = \frac{x}{2} \cdot \frac{1}{2y} $. The common factor is $ \frac{x}{2} $.

Q: Why can't I factor out just x instead of x/2?

If you factor out just x, you get $ x(\frac{y}{2} + \frac{1}{4y}) $, which is correct but not fully factored. Factoring out $ \frac{x}{2} $ gives the most simplified form.

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

Use the distributive property to expand your factored form. If you get back to the original expression, your factoring is correct!

Q: Why is factoring rational expressions important?

Factoring helps simplify complex expressions, solve equations more easily, and identify key features like zeros and asymptotes in advanced mathematics.

Algebraic Factoring with Rational Expressions

Decompose the following expression into factors:

$\frac{xy}{2}+\frac{x}{4y}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's find a common factor.

00:14 We can break down four into two and two.

00:27 Now, mark the common factors.

00:36 Take these common factors out of the parentheses.

00:46 And that's how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Decompose the following expression into factors:

$\frac{xy}{2}+\frac{x}{4y}$

Step-by-step solution

To factor the expression $\frac{xy}{2}+\frac{x}{4y}$ , we proceed as follows:

Step 1: Identify the common factor between the terms.
Step 2: Factor out the common factor.
Step 3: Simplify the expression inside the parentheses.

Let's break this down:
Step 1: The expression is $\frac{xy}{2} + \frac{x}{4y}$ . Clearly, both terms share $\frac{x}{2}$ as a common factor.
Step 2: Factor out $\frac{x}{2}$ from each term:
- From the first term: $\frac{xy}{2} = \frac{x}{2} \times y$ .
- From the second term: $\frac{x}{4y} = \frac{x}{2} \times \frac{1}{2y}$ .
Step 3: This gives us:
$\frac{xy}{2} + \frac{x}{4y} = \frac{x}{2}(y + \frac{1}{2y})$

Thus, the expression can be decomposed into factors as $\frac{x}{2}(y+\frac{1}{2y})$ .

Final Answer

$\frac{x}{2}(y+\frac{1}{2y})$

Key Points to Remember

Essential concepts to master this topic

Common Factor: Look for shared variables and coefficients in all terms
Technique: Factor out $\frac{x}{2}$ from both $\frac{xy}{2}$ and $\frac{x}{4y}$
Check: Distribute back: $\frac{x}{2} \cdot y + \frac{x}{2} \cdot \frac{1}{2y} = \frac{xy}{2} + \frac{x}{4y}$ ✓

Common Mistakes

Avoid these frequent errors

Factoring out only the variable without considering denominators
Don't just factor out x and ignore the denominators = wrong simplification! This leaves fractions that can't be properly factored. Always identify the complete common factor including both numerator and denominator parts.

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 common factor when terms have different denominators?

Look at each term separately: $\frac{xy}{2} = \frac{x}{2} \cdot y$ and $\frac{x}{4y} = \frac{x}{2} \cdot \frac{1}{2y}$ . The common factor is $\frac{x}{2}$ .

Why can't I factor out just x instead of x/2?

If you factor out just x, you get $x(\frac{y}{2} + \frac{1}{4y})$ , which is correct but not fully factored. Factoring out $\frac{x}{2}$ gives the most simplified form.

How do I check if my factoring is correct?

Use the distributive property to expand your factored form. If you get back to the original expression, your factoring is correct!

What if the terms don't seem to have a common factor?

Look more carefully! In rational expressions, the common factor might include fractions. Write each term as a product to see the common parts clearly.

Can I factor this expression differently?

While there might be other ways to group terms, factoring out the greatest common factor gives you the most simplified and useful form.

Why is factoring rational expressions important?

Factoring helps simplify complex expressions, solve equations more easily, and identify key features like zeros and asymptotes in advanced mathematics.

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