5.1 Rational expressions & equations

What Is a Rational Expression?

A rational expression is a fraction whose numerator and denominator are both polynomials, such as (x^2 - 9)/(x + 3). Because you can never divide by zero, every rational expression carries excluded values — the numbers that make the denominator equal zero. Finding those values is the first move on almost every PERT rational-expression question, so train yourself to look at the denominator before you do anything else.

Finding Excluded Values

Set the denominator equal to zero and solve. For (x + 5)/(x^2 - 4), factor the bottom: x^2 - 4 = (x - 2)(x + 2). The denominator is zero when x = 2 or x = -2, so both numbers are excluded. The expression is undefined there even when the numerator behaves perfectly well. A common mistake is to factor carelessly, so take the time to factor fully, because a complete factorization is what reveals every value the expression forbids.

Simplifying by Factoring and Canceling

To simplify, factor the numerator and denominator completely, then cancel any factor they share.

Example 1. Simplify (x^2 - 9)/(x^2 + 7x + 12). Factor each piece: x^2 - 9 = (x - 3)(x + 3), and x^2 + 7x + 12 = (x + 3)(x + 4). So the expression becomes [(x - 3)(x + 3)] / [(x + 3)(x + 4)]. Cancel the shared (x + 3) factor, leaving (x - 3)/(x + 4), with x = -3 and x = -4 still excluded.

Only whole factors cancel — never cross out individual terms. You cannot cancel the x's in (x + 2)/x, because that x is added, not multiplied.

Multiplying and Dividing

Multiply rational expressions the way you multiply numeric fractions: factor everything, cancel across the two fractions, then multiply straight across.

Example 2. Multiply (x^2 - 1)/(x + 4) times (x + 4)/(x - 1). Factor x^2 - 1 = (x - 1)(x + 1), giving [(x - 1)(x + 1)]/(x + 4) times (x + 4)/(x - 1). Cancel (x + 4) and (x - 1); what remains is (x + 1)/1 = x + 1.

To divide, multiply by the reciprocal — flip the second fraction — then proceed as a product.

Example 3. Divide (3x)/(x + 2) by (9x^2)/(x^2 - 4). Flip the divisor: (3x)/(x + 2) times (x^2 - 4)/(9x^2). Factor x^2 - 4 = (x - 2)(x + 2). = (3x)/(x + 2) times [(x - 2)(x + 2)]/(9x^2). Cancel (x + 2), and reduce 3x/(9x^2) to 1/(3x). The final answer is (x - 2)/(3x).

Adding and Subtracting

Addition and subtraction need a common denominator, exactly like numeric fractions.

• Same denominator: combine the numerators and keep the denominator.
• Different denominators: build the least common denominator (LCD), rewrite each fraction, then combine.

Example 4. Add 2/x + 3/(x + 1). The LCD is x(x + 1). Rewrite each fraction: 2(x + 1)/[x(x + 1)] + 3x/[x(x + 1)]. Combine the numerators: [2(x + 1) + 3x] / [x(x + 1)] = (2x + 2 + 3x)/[x(x + 1)] = (5x + 2)/[x(x + 1)].

Example 5. Subtract (x + 3)/(x - 2) - 4/(x - 2). The denominators match, so combine numerators — and distribute the minus sign carefully: (x + 3 - 4)/(x - 2) = (x - 1)/(x - 2).

Solving Rational Equations

A rational equation sets two expressions equal, so you may clear the fractions entirely instead of just combining them.

Method: multiply every term by the LCD, solve the resulting polynomial equation, then check each answer against the excluded values. Any solution that makes an original denominator zero is extraneous and must be discarded.

Example 6. Solve 1/x + 1/2 = 3/x. The LCD is 2x. Multiply every term: 2x(1/x) + 2x(1/2) = 2x(3/x), which gives 2 + x = 6. So x = 4. Since x = 4 makes no denominator zero, it is a valid solution.

Example 7 (cross-multiplying). Solve 5/(x - 1) = 10/(x + 3). With a single fraction on each side you can cross-multiply: 5(x + 3) = 10(x - 1), so 5x + 15 = 10x - 10, then 25 = 5x, so x = 5. Neither denominator is zero at x = 5, so x = 5 is the solution.

Example 8 (extraneous). Solve x/(x - 3) = 3/(x - 3) + 2. Multiply by (x - 3): x = 3 + 2(x - 3), so x = 3 + 2x - 6, then x = 2x - 3, giving x = 3. But x = 3 makes the denominator zero, so it is extraneous — the equation has no solution.

Quick Reference

Always state excluded values, and always test your solutions — on the PERT a "no solution" answer choice is frequently there to catch an extraneous root you forgot to check.