An algebraic fraction is a fraction that has variables (letters) in the numerator, denominator, or both.

Examples:

x / 5, 3x+2 / 7, 2x / x+4, x^2−1 / x​

Just like regular fractions, algebraic fractions represent division.

So:

2x / x+4 = “2x÷(x+4)”

⭐ Parts of an Algebraic Fraction

Example:

3x+1 / x−2​

• Numerator (top): $3x+1$

• Denominator (bottom): $x-2$

📌 Important rule: The denominator can never be 0
So for this fraction:

x−2≠0⇒x≠2

That’s called a restriction (or “excluded value”).

✅ Simplifying Algebraic Fractions

Simplifying algebraic fractions is like simplifying regular fractions:

Key rule:

You can only cancel factors, not terms.

✅ Canceling factors (allowed)

6x / 3 = 2x

❌ Canceling terms (not allowed)

x+2 / x ≠ 2 / 1​

You cannot cancel the x across addition.

🔥 Example: Simplify

6x / 9​

Divide top and bottom by 3:

2x / 3​

🔥 Example with factoring

x2−9 / x+3​

Step 1: Factor the numerator:

x^2 – 9 = (x-3)(x+3)

Now:

(x−3)(x+3) / x+3​

Cancel the common factor $x+3$:

$$x-3$$

✅ Simplified result:

$$x-3$$

📌 BUT restriction still matters:

x≠−3x 

Even though it cancels, the original fraction would be undefined at x = -3

✅ Multiplying Algebraic Fractions

Multiply straight across:

a / b ⋅ c / d = ac / bd​

Example:

2x / 3 ⋅ 9 / x = 18x / 3x​

Cancel:

$$=6$$

(Restriction: x≠0)

✅ Dividing Algebraic Fractions

Dividing means multiply by the reciprocal:

a / b ÷ c / d = a / b ⋅ d / c​

Example:

x / 4 ÷ 2 / x = x / 4 ⋅ x / 2 = x^2 / 8​

(Restriction: x ≠ 0)

✅ Adding/Subtracting Algebraic Fractions (Hardest Part)

Just like regular fractions, you need a common denominator.

Example:

1 / x   +   2 / x = 3 / x​

Easy because denominators match.

Example with different denominators:

1 / x     +     1 / x+1​

Common denominator:

$$x(x+1)$$

Rewrite:

x+1 / x(x+1)  +   x / x(x+1)​

Add:

2x+1 / x(x+1)​