The students were asked to add or subtract the given...

The students were asked to add or subtract the given rational expressions. Explain what the student did wrong. Give the correct answers. $\frac{b+3}{b+5}-\frac{5 b+2}{b+5}$ $$ \begin{aligned} &\text { Hung }\\ &\begin{aligned} \frac{b+3}{b+5}-\frac{5 b+2}{b+5} &=\frac{b+3-5 b+2}{b+5} \\ &=\frac{-4 b+5}{b+5} \end{aligned} \end{aligned} $$

The students were asked to add or subtract the given rational expressions. Explain what the student did wrong. Give the correct answers.
$\frac{b+3}{b+5}-\frac{5 b+2}{b+5}$
$$
\begin{aligned}
&\text { Hung }\\
&\begin{aligned}
\frac{b+3}{b+5}-\frac{5 b+2}{b+5} &=\frac{b+3-5 b+2}{b+5} \\
&=\frac{-4 b+5}{b+5}
\end{aligned}
\end{aligned}
$$

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Key Concepts

Common Denominator

When adding or subtracting rational expressions, it is essential to have a common denominator. This process allows you to combine the numerators directly while keeping the denominator intact, mirroring the methodology used in adding or subtracting numerical fractions.

Combining Like Terms

After rewriting the expression with a common denominator, the next step is to combine like terms in the numerator. This involves carefully adding or subtracting the coefficients of identical power terms, an important process in simplifying polynomial expressions.

Rational Expressions

Rational expressions are quotients of polynomials. They behave similarly to numerical fractions and require careful attention to the rules for arithmetic operations, like addition and subtraction, particularly with regard to common denominators and restrictions on the variables.

Arithmetic Operations on Polynomials

Performing operations on polynomials—including addition, subtraction, and especially handling negative signs—requires precision. A minor arithmetic mistake, such as misaligning signs during subtraction, can lead to incorrect results, emphasizing the need for careful distribution of negative signs and accurate combination of constants and coefficients.

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