Numerator of the First Fraction

Oct 13, 2025 by talynt.org 7 views

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Introduction

In this article, we will simplify the given expression using algebraic techniques. The expression involves fractions and division, which can be simplified using various methods. We will break down the expression into smaller parts and simplify each part step by step.

The Given Expression

The given expression is:

\frac{c^2-c-20}{c^2-6c+5} \div \frac{c^2-16}{3c-3}

Step 1: Factorize the Numerator and Denominator

To simplify the expression, we need to factorize the numerator and denominator of each fraction.

Numerator of the First Fraction

The numerator of the first fraction is $c^2-c-20$ . We can factorize it as:

c^2-c-20 = (c-5)(c+4)

Denominator of the First Fraction

The denominator of the first fraction is $c^2-6c+5$ . We can factorize it as:

c^2-6c+5 = (c-5)(c-1)

Numerator of the Second Fraction

The numerator of the second fraction is $c^2-16$ . We can factorize it as:

c^2-16 = (c-4)(c+4)

Denominator of the Second Fraction

The denominator of the second fraction is $3c-3$ . We can factorize it as:

3c-3 = 3(c-1)

Step 2: Simplify the Expression

Now that we have factorized the numerator and denominator of each fraction, we can simplify the expression.

Simplify the First Fraction

The first fraction is:

\frac{(c-5)(c+4)}{(c-5)(c-1)}

We can cancel out the common factor $(c-5)$ from the numerator and denominator:

\frac{(c-5)(c+4)}{(c-5)(c-1)} = \frac{c+4}{c-1}

Simplify the Second Fraction

The second fraction is:

\frac{(c-4)(c+4)}{3(c-1)}

We can cancel out the common factor $(c-1)$ from the numerator and denominator:

\frac{(c-4)(c+4)}{3(c-1)} = \frac{(c-4)(c+4)}{3(c-1)}

Step 3: Divide the Fractions

Now that we have simplified the fractions, we can divide them.

Divide the Fractions

The expression is:

\frac{c+4}{c-1} \div \frac{(c-4)(c+4)}{3(c-1)}

We can invert the second fraction and multiply:

\frac{c+4}{c-1} \div \frac{(c-4)(c+4)}{3(c-1)} = \frac{c+4}{c-1} \times \frac{3(c-1)}{(c-4)(c+4)}

We can cancel out the common factors $(c-1)$ and $(c+4)$ :

\frac{c+4}{c-1} \times \frac{3(c-1)}{(c-4)(c+4)} = \frac{3}{c-4}

Conclusion

In this article, we simplified the given expression using algebraic techniques. We factorized the numerator and denominator of each fraction, simplified the fractions, and divided them. The final simplified expression is:

\frac{3}{c-4}

This expression matches option A.

Discussion

The given expression involves fractions and division, which can be simplified using various methods. We used algebraic techniques to simplify the expression, including factorization and cancellation of common factors. The final simplified expression is a fraction with a constant numerator and a linear denominator.

Final Answer

The final answer is:

\boxed{\frac{3}{c-4}}$$<br/> # Simplify the Expression: A Q&A Guide =====================================================

Introduction

In our previous article, we simplified the given expression using algebraic techniques. In this article, we will provide a Q&A guide to help you understand the steps involved in simplifying the expression.

Q: What is the given expression?

A: The given expression is: