Rational Expressions and Equations

A rational expression is a fraction where the numerator and/or denominator is a polynomial, such as (x² - 4)/(x + 2). Working with rational expressions requires strong factoring skills.

Simplifying means canceling common factors (not terms). (x² - 4)/(x + 2) = (x+2)(x-2)/(x+2) = x - 2, provided x ≠ -2. Always state restrictions where the denominator equals zero.

Multiplying and dividing rational expressions:

• Multiply: multiply numerators, multiply denominators, then simplify
• Divide: multiply by the reciprocal of the second fraction

Adding and subtracting requires a common denominator. Find the LCD, rewrite each fraction, then add/subtract numerators. Example: 1/x + 2/(x+1) = (x+1)/(x(x+1)) + 2x/(x(x+1)) = (3x+1)/(x(x+1)).

Solving rational equations — multiply through by the LCD to eliminate all fractions, then solve the resulting polynomial equation.

Extraneous solutions arise when a solution makes the original denominator zero. Always substitute answers back to verify. For example, if x = 2 appears as a solution but the original had (x - 2) in the denominator, discard it.

Key restrictions to state:

• Denominator ≠ 0
• Under a square root ≥ 0 (if applicable)

⚠️ Common mistake: canceling terms (not factors). (x + 3)/(x + 5) cannot be simplified by canceling the x's.

💡 Tip: On the SAT®, when you solve a rational equation and get two answers, always check both to see if either is extraneous.