Rational expressions

Fractions with polynomials

A rational expression is a fraction with polynomials on the top and bottom:

• (x + 3)/(x − 2)
• (x² − 9)/(x² + 5x + 6)
• 4/x

Restrictions

Division by zero is undefined, so any value that makes the denominator zero is a restriction.

Example: (x + 1)/(x² − 4)

• x² − 4 = (x + 2)(x − 2) = 0 when x = 2 or x = −2
• Restrictions: x ≠ 2, x ≠ −2

Always find restrictions from the original denominator, before cancelling.

Simplifying

To simplify:

1. Factor the numerator and denominator completely.
2. State restrictions.
3. Cancel common factors (not terms).

Example: (x² − 9)/(x² + 5x + 6)

• = (x + 3)(x − 3) / [(x + 3)(x + 2)]
• Restrictions: x ≠ −3, −2
• = (x − 3)/(x + 2), x ≠ −3, −2

The result is in lowest terms.

Cancel factors, never terms

• (x + 5)/5 ≠ x. The 5 is a term on top, not a factor.
• (5x)/5 = x ✓ (5 is a factor of 5x)

Opposite factors

(a − b) = −(b − a), so they cancel to −1.

Example: (3 − x)/(x − 3) = −1, x ≠ 3

More examples

Why restrictions matter

(x² − 9)/(x − 3) simplifies to x + 3, but the original is undefined at x = 3. Its graph is the line y = x + 3 with a hole at (3, 6).

Worked examples

Example 1: Simplify (5x + 10)/(x² − 4). → 5(x + 2)/[(x + 2)(x − 2)] = 5/(x − 2), x ≠ ±2.

Example 2: Restrictions on 7/(x² + 3x)? x(x + 3) → x ≠ 0, −3.

Example 3: Simplify (x² − 2x − 15)/(x² − 25). → (x − 5)(x + 3)/[(x − 5)(x + 5)] = (x + 3)/(x + 5), x ≠ ±5.

Common mistakes

• Cancelling terms instead of factors.
• Finding restrictions after cancelling.
• Not factoring completely.

Try it yourself

1. Restrictions on (x − 1)/(x² − x − 6)?
2. Simplify 12x³/(4x).
3. Simplify (x² − 1)/(x² + 2x + 1).
4. Simplify (4 − x)/(x² − 16).

Answers: 1) x ≠ 3, −2 2) 3x², x ≠ 0 3) (x − 1)/(x + 1), x ≠ −1 4) −1/(x + 4), x ≠ ±4

Quick recap

• A rational expression is a fraction of polynomials.
• State restrictions from the original denominator.
• Factor, then cancel common factors to simplify.