Understanding Rational Expressions

A rational expression is simply a fraction where both parts are polynomials, written as $\frac{p}{q}$ where $q ≠ 0$. When evaluating these expressions, just substitute the variable value and calculate.

Before working with rational expressions, you need to identify restricted values—values that make the denominator zero (which would cause division by zero). To find these, set the denominator equal to zero and solve. For example, in $\frac{y-3}{2y+7}$, setting $2y+7=0$ gives us $y = -\frac{7}{2}$ as the restricted value.

When simplifying rational expressions, factor both numerator and denominator completely, then cancel common factors. Like in $\frac{2p-14}{p^2-49}$, we can rewrite this as $\frac{2(p-7)}{(p+7)(p-7)}$, which simplifies to $\frac{2}{p+7}$ since the $(p-7)$ terms cancel.

Pro Tip: Always identify restricted values before simplifying! This prevents losing critical information about where the expression is undefined.

Simplifying and Multiplying Rational Expressions

Simplifying rational expressions often requires factoring skills. For instance, $\frac{5-y}{y^2-25}$ can be rewritten as $\frac{-(y-5)}{(y+5)(y-5)}$, which simplifies to $\frac{-1}{y+5}$. Notice how recognizing the difference of squares pattern in the denominator was key!

When multiplying rational expressions, follow these steps:

1. Factor all numerators and denominators completely
2. Cancel any common factors between numerators and denominators
3. Multiply the remaining factors in the numerators and denominators separately

For example, when multiplying $\frac{3c-3d}{6c} \cdot \frac{2}{c^2-d^2}$, first rewrite as $\frac{3(c-d)}{6c} \cdot \frac{2}{(c-d)(c+d)}$. The common factor $(c-d)$ cancels, leaving $\frac{1}{c} \cdot \frac{1}{c+d} = \frac{1}{c(c+d)}$.

Remember: The formula for multiplying rational expressions is $\frac{p}{q} \cdot \frac{r}{s} = \frac{pr}{qs}$ – but only after you've factored and simplified!

Division of Rational Expressions

Division with rational expressions isn't as scary as it seems! The key trick is to convert division into multiplication by using the reciprocal of the divisor.

To divide rational expressions, flip (take the reciprocal of) the second fraction and change the division to multiplication. For example, $\frac{5+15}{2} \div \frac{+2-9}{10}$ becomes $\frac{5+15}{2} \cdot \frac{10}{+2-9}$.

After converting to multiplication, follow the same process: factor completely, cancel common factors, then multiply remaining terms. This works for complex expressions too!

When working with multiple divisions like $\frac{3x}{4y} \div \frac{3x}{4y} \div \frac{5x}{6y}$, handle them one at a time, working from left to right. Converting each division to multiplication makes these problems much more manageable.

Quick Tip: Division is just multiplication by the reciprocal! Always flip the second fraction and change ÷ to ×.

Addition and Subtraction of Rational Expressions

Adding or subtracting rational expressions is easiest when the denominators are the same. With identical denominators, you can simply combine the numerators while keeping the denominator: $\frac{p}{q} + \frac{r}{q} = \frac{p+r}{q}$.

For example, $\frac{2}{3d+5} + \frac{7d}{3d+5} = \frac{2+7d}{3d+5}$ or $\frac{7d+2}{3d+5}$. Notice that you don't cancel anything here—the numerator and denominator don't share common factors.

When denominators differ, you'll need to find the least common denominator (LCD). The LCD contains all factors from all denominators, each raised to its highest occurring power. For fractions like $\frac{5}{14}$, $\frac{3}{49}$, and $\frac{1}{8}$, factor the denominators first: $\frac{5}{2 \cdot 7}$, $\frac{3}{7^2}$, $\frac{1}{2^3}$. The LCD would be $7^2 \cdot 2^3 = 392$.

Heads Up: When adding or subtracting rational expressions, don't try to cancel terms in the numerator with terms in the denominator! This only works with multiplication and division.