Factorize the Expression: xy/8 + xy²/16 + xy³/20 Step-by-Step

Q: Why can't I just factor out xy and be done?

While $ xy $ is a common factor, you can simplify further! The denominators 8, 16, and 20 also have common factors. Complete factorization means finding all possible common factors.

Q: How do I find the GCD of 8, 16, and 20?

List the factors: 8 = 2³, 16 = 2⁴, 20 = 2² × 5. The GCD is 4 (the highest power of 2 that divides all three). So you can factor out $ \frac{1}{4} $ from the denominators.

Q: What does the final factored form mean?

$ \frac{xy}{4}\left(\frac{1}{2}+\frac{y}{4}+\frac{y^2}{5}\right) $ means xy divided by 4 times a polynomial in y. This shows the structure of the original expression more clearly.

Q: How can I check if my factorization is correct?

Distribute the factored form back out! Multiply $ \frac{xy}{4} $ by each term inside the parentheses. If you get back to $ \frac{xy}{8}+\frac{xy^2}{16}+\frac{xy^3}{20} $, you're correct!

Q: Why is this form better than the original?

Factored form reveals common structure and makes it easier to work with the expression in further calculations. It also shows clearly what happens when x = 0 or y = 0.

Factoring Algebraic Expressions with Mixed Denominators

Decompose the following expression into factors:

$\frac{xy}{8}+\frac{xy^2}{16}+\frac{xy^3}{20}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Break into factors

00:05 Let's factor 8 into 4 and 2

00:10 Let's break down the square into products

00:13 Let's factor 16 into 4 and 4

00:22 Let's break down the power of 3 into square and product

00:25 Let's factor 20 into 4 and 5

00:28 Let's mark the common factors

00:51 Let's take out the common factors from the parentheses

01:04 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

Decompose the following expression into factors:

$\frac{xy}{8}+\frac{xy^2}{16}+\frac{xy^3}{20}$

Step-by-step solution

To factor the given expression, we proceed as follows:

Step 1: Identify Common Factors
The terms $\frac{xy}{8}$ , $\frac{xy^2}{16}$ , and $\frac{xy^3}{20}$ have $xy$ as a common factor in the numerators.
Step 2: Factor Out Common Numerator
Factor $xy$ out of each term:
$\frac{xy}{8} = \frac{xy}{8}$ ,
$\frac{xy^2}{16} = \frac{xy \cdot y}{16}$ ,
$\frac{xy^3}{20} = \frac{xy \cdot y^2}{20}$ .
Step 3: Simplify the Expression
Factor out $xy$ and adjust fractions:
$xy \left( \frac{1}{8} + \frac{y}{16} + \frac{y^2}{20} \right)$ .
Step 4: Simplify Denominator Terms
Factor $\frac{1}{4}$ common to $\frac{1}{8}$ , $\frac{y}{16}$ , and $\frac{y^2}{20}$ :
$\frac{xy}{4} \left( \frac{1}{2} + \frac{y}{4} + \frac{y^2}{5} \right)$ .

Thus, the expression $\frac{xy}{8} + \frac{xy^2}{16} + \frac{xy^3}{20}$ decomposes to the factored form $\frac{xy}{4} \left( \frac{1}{2} + \frac{y}{4} + \frac{y^2}{5} \right)$ .

Final Answer

$\frac{xy}{4}(\frac{1}{2}+\frac{y}{4}+\frac{y^2}{5})$

Key Points to Remember

Essential concepts to master this topic

Common Factors: Look for variables and coefficients shared by all terms
Technique: Factor out $xy$ first, then find GCD of denominators 8, 16, 20
Check: Expand your factored form to verify it equals the original expression ✓

Common Mistakes

Avoid these frequent errors

Only factoring out variables without considering denominators
Don't just factor $xy$ and stop there = incomplete factorization! This misses the opportunity to simplify the fraction coefficients further. Always look for common factors in both numerators AND denominators to get the most simplified form.

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

Why can't I just factor out xy and be done?

While $xy$ is a common factor, you can simplify further! The denominators 8, 16, and 20 also have common factors. Complete factorization means finding all possible common factors.

How do I find the GCD of 8, 16, and 20?

List the factors: 8 = 2³, 16 = 2⁴, 20 = 2² × 5. The GCD is 4 (the highest power of 2 that divides all three). So you can factor out $\frac{1}{4}$ from the denominators.

What does the final factored form mean?

$\frac{xy}{4}\left(\frac{1}{2}+\frac{y}{4}+\frac{y^2}{5}\right)$ means xy divided by 4 times a polynomial in y. This shows the structure of the original expression more clearly.

How can I check if my factorization is correct?

Distribute the factored form back out! Multiply $\frac{xy}{4}$ by each term inside the parentheses. If you get back to $\frac{xy}{8}+\frac{xy^2}{16}+\frac{xy^3}{20}$ , you're correct!

Why is this form better than the original?

Factored form reveals common structure and makes it easier to work with the expression in further calculations. It also shows clearly what happens when x = 0 or y = 0.

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