Lesson 37: Adding and Subtracting Rational Expressions

New Concept

A rational expression is a ratio of polynomials: $\frac{f(x)}{g(x)}$, for polynomials $f(x)$ and $g(x)$, where $g(x) \neq 0$ and is of degree $\geq 1$.

What’s next

Next, you’ll practice adding and subtracting these expressions by finding a common denominator, just like with numerical fractions.

When adding or subtracting rational expressions with the same denominator, you simply combine the numerators and place the result over the common denominator. The rule is $\frac{A}{C} + \frac{B}{C} = \frac{A+B}{C}$. Be careful to distribute the negative sign when subtracting numerators that are polynomials.

$\frac{5}{4y} + \frac{7}{4y} = \frac{5+7}{4y} = \frac{12}{4y} = \frac{3}{y}$
$\frac{z}{z^2 - 16} - \frac{4}{z^2 - 16} = \frac{z-4}{(z-4)(z+4)} = \frac{1}{z+4}$
$\frac{3a}{a^2+a-6} - \frac{a-4}{a^2+a-6} = \frac{3a-(a-4)}{(a+3)(a-2)} = \frac{2a+4}{(a+3)(a-2)}$

Think of it like adding pizza slices! If you have $\frac{1}{8}$ of a pizza and someone gives you $\frac{3}{8}$ more, you just add the numerators to find out how many slices you have in total. The size of the slices, which is the denominator, stays the same. The same simple logic applies to algebraic fractions!

The least common denominator (LCD) is the least common multiple (LCM) of the denominators of the fractions. To find the LCD of polynomial denominators, first factor each polynomial completely. Then, construct the LCD by taking the highest power of each unique factor that appears in any of the denominators and multiplying them together.

For $\frac{1}{4x^3}$ and $\frac{1}{6x^2}$, the factors are $2^2, x^3$ and $2, 3, x^2$. The LCD is $2^2 \cdot 3 \cdot x^3 = 12x^3$.
For $\frac{1}{x^2-4}$ and $\frac{1}{x^2+x-6}$, factor to $\frac{1}{(x-2)(x+2)}$ and $\frac{1}{(x+3)(x-2)}$. The LCD is $(x-2)(x+2)(x+3)$.

Think of the LCD as finding a 'common language' for different fractions before they can be combined. By breaking down each denominator into its prime factors, you can see all the essential 'building blocks.' The LCD is the smallest and most efficient denominator that includes every single one of these blocks, ensuring all fractions can be converted smoothly.

A rational expression in the form $\frac{f(x)}{g(x)}$ is undefined for any value of the variable that causes the denominator, $g(x)$, to equal zero. This is because division by zero is not a defined operation in mathematics. To find these values, you must set the denominator equal to zero and solve for the variable.

The expression $\frac{x+2}{4x-12}$ is undefined when $4x-12=0$, which happens when $x=3$.
The expression $\frac{3x}{x^2-25}$ is undefined when $x^2-25=0$. Factoring gives $(x-5)(x+5)=0$, so it is undefined at $x=5$ and $x=-5$.

Remember the most important rule in fractions: you can never, ever divide by zero! It's a mathematical impossibility. Finding the 'undefined values' is like finding the secret trap doors in your equation. By setting the denominator to zero, you are identifying the specific inputs that would break the math, helping you steer clear of trouble.