Decompose the Expression: 18ab/c² - 99c²/ab

Q: How do I know what to factor out of rational expressions?

Look for the greatest common factor in both the coefficients and the variables. Here, we have 9 as the GCD of 18 and 99, and $ \frac{ab}{c^2} $ appears in a form in both terms.

Q: Why does the second term become such a complicated fraction?

When we factor out $ \frac{ab}{c^2} $ from $ -99\frac{c^2}{ab} $, we get $ -11\frac{c^4}{a^2b^2} $ because dividing fractions means multiplying by the reciprocal!

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

Distribute everything back out! Multiply $ 9\frac{ab}{c^2} $ by each term inside the parentheses. You should get exactly the original expression.

Q: What if none of the answer choices look right?

Double-check your arithmetic, especially when handling negative signs and complex fractions. Make sure you factored out the common factor correctly from both terms.

Q: Is there a pattern for factoring rational expressions?

Find the GCD of coefficientsIdentify the common algebraic factorFactor both out togetherAlways verify by expanding back!

Algebraic Factoring with Rational Expressions

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

$18\frac{ab}{c^2}-99\frac{c^2}{ab}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find a common factor

00:03 Factor 18 into factors 9 and 2

00:09 Factor 99 into factors 11 and 9

00:13 Mark the common factors

00:22 Take out the common factors from the parentheses

00:46 Mark the common factors

00:56 Take out the common factors from the parentheses

01:08 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 expression below?

$18\frac{ab}{c^2}-99\frac{c^2}{ab}$

Step-by-step solution

To solve this problem, we will follow these steps:

Step 1: Determine the greatest common divisor (GCD) of the numeric coefficients 18 and 99.
The GCD of 18 and 99 is 9 since $18 = 2 \times 9$ and $99 = 11 \times 9$ .
Step 2: Analyze the variables in the terms $18\frac{ab}{c^2}$ and $-99\frac{c^2}{ab}$ .
Both terms have the fraction form with $\frac{ab}{c^2}$ and $-\frac{c^2}{ab}$ .
Step 3: Identify the common algebraic fraction factor.
The common factor from the algebraic component is $\frac{ab}{c^2}$ .
Step 4: Factor $\frac{ab}{c^2}$ out from the expression.
The expression $18\frac{ab}{c^2} - 99\frac{c^2}{ab}$ can be rewritten as:
$9\frac{ab}{c^2}(2 - 11\frac{c^4}{a^2b^2})$ .

Therefore, the decomposition of the given expression is $9\frac{ab}{c^2}(2 - 11\frac{c^4}{a^2b^2})$ .

Final Answer

$9\frac{ab}{c^2}(2-11\frac{c^4}{a^2b^2})$

Key Points to Remember

Essential concepts to master this topic

GCD First: Find greatest common divisor of coefficients (18 and 99 = 9)
Common Factor: Extract $\frac{ab}{c^2}$ from both terms to simplify
Check: Expand your factored form and verify it equals the original expression ✓

Common Mistakes

Avoid these frequent errors

Factoring out the wrong common factor
Don't just factor out the GCD of coefficients (9) without considering the algebraic parts = incomplete factorization! This leaves complex fractions that don't match any answer choice. Always identify both the numerical GCD and the common algebraic factor 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 know what to factor out of rational expressions?

Look for the greatest common factor in both the coefficients and the variables. Here, we have 9 as the GCD of 18 and 99, and $\frac{ab}{c^2}$ appears in a form in both terms.

Why does the second term become such a complicated fraction?

When we factor out $\frac{ab}{c^2}$ from $-99\frac{c^2}{ab}$ , we get $-11\frac{c^4}{a^2b^2}$ because dividing fractions means multiplying by the reciprocal!

How can I check if my factorization is correct?

Distribute everything back out! Multiply $9\frac{ab}{c^2}$ by each term inside the parentheses. You should get exactly the original expression.

What if none of the answer choices look right?

Double-check your arithmetic, especially when handling negative signs and complex fractions. Make sure you factored out the common factor correctly from both terms.

Is there a pattern for factoring rational expressions?

Find the GCD of coefficients
Identify the common algebraic factor
Factor both out together
Always verify by expanding back!

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