91Ó°ÊÓ

When working with rational expressions, factoring polynomials is a crucial step. Factoring simplifies the expressions and makes it easier to identify common denominators. In our problem, we start by factoring the original denominator, which is \(5r - 5\).
To factor this, we notice that both terms share a common factor of 5. We can then factor out the 5:

• $$5r - 5 = 5(r - 1)$$

This simplification helps us to see the relation with the indicated denominator, \(15r - 15\).
Similar to the first step, we factor the indicated denominator:

• $$15r - 15 = 15(r - 1)$$

By factoring polynomials, we break down the expressions into simpler, recognizable factors, which is essential for further steps.

Finding common denominators is a key part of working with rational expressions. In our exercise, we need to rewrite the original fraction with a different denominator. Once we have factored both the original and indicated denominators, we can easily identify the relationship.
The original and indicated denominators after factoring are:

• Original: $$5(r - 1)$$
• Indicated: $$15(r - 1)$$

Here, both denominators have the common factor \((r - 1)\). The original denominator needs to be multiplied by a certain factor to match the indicated denominator.
We notice that 5 must be multiplied by 3 to get 15:

• $$5 \times 3 = 15$$

This multiplication factor is applied to both the numerator and the denominator to convert the original fraction to one with a common denominator.

Understanding how fractions work is essential when reworking rational expressions. In this problem, we rewrite the original fraction \(\frac{3r}{5r - 5}\) with a new denominator \(15r - 15\).
First, apply the factor of 3 to both the numerator and denominator:

• Numerator: $$3r \times 3 = 9r$$
• Denominator: $$5(r - 1) \times 3 = 15(r - 1)$$

By multiplying both parts by the factor determined previously, we keep the value of the fraction unchanged. The final expression is:

• $$\frac{9r}{15r - 15}$$

This way, we have successfully rewritten the fraction with the indicated denominator.
Understanding fractions and multiplication factors is vital in making these conversions seamlessly.