Math 20-1 · Rational expressions
Rational expressions and equations
A rational expression is a fraction with algebra in it, and every rule you already know about fractions still applies. What is new is that the denominator can become zero, and you have to say where.
- 1. The words, first
- 2. Non-permissible values
- 3. Multiplying, dividing, adding and subtracting
- 4. Rational equations
- 5. What costs marks
The words, first
The idea: One new idea — the non-permissible value — and a lot of old fraction rules wearing new clothes.
| Word | What it means |
|---|---|
| Rational expression | A quotient of two polynomials, like (x + 2)/(x − 5). Rational means ratio, not sensible. |
| Non-permissible value | A value of the variable that makes a denominator zero, so the expression is undefined there. Often shortened to NPV. |
| Factored form | The expression written as products, so common factors are visible. Nothing cancels until everything is factored. |
| Simplify | Divide out factors common to the numerator and denominator — and carry the restrictions forward. |
| Lowest common denominator | The smallest expression every denominator divides into. Built from the factored denominators, taking each distinct factor the greatest number of times it appears. |
| Reciprocal | The fraction turned upside down. Dividing by a fraction is multiplying by its reciprocal. |
| Rational equation | An equation containing rational expressions. Multiplying through by the LCD clears the fractions. |
Non-permissible values
The idea: Division by zero is undefined, so before anything else, find every value that makes a denominator zero and write it down.
Find them from the factored denominator. For x/(x² − 4), factor to x/[(x − 2)(x + 2)], so x ≠ 2 and x ≠ −2. A quadratic denominator usually produces two restrictions, and you will miss one if you do not factor.
Restrictions survive cancelling. Look at (x² − 9)/(x + 3) = (x − 3)(x + 3)/(x + 3) = x − 3. The simplified expression looks fine at x = −3, but the original was undefined there, and the two are only equal where both are defined. The correct answer is x − 3, x ≠ −3. Dropping that restriction is the single most common error in this unit.
Division has extra restrictions. When you divide, the numerator of the divisor also ends up in a denominator, so it cannot be zero either. For (x² − 1)/(x + 2) ÷ (x − 1)/(x + 2), the restrictions are x ≠ −2 and x ≠ 1.
Multiplying, dividing, adding and subtracting
The idea: Factor first, every time. After that the rules are the ones you learned with numbers.
Multiply straight across, then cancel: (x/3)(6/x²) = 6x/3x² = 2/x.
Divide by multiplying by the reciprocal, and only then cancel.
Add and subtract over a common denominator. To do 1/x + 1/(x + 1), the LCD is x(x + 1):
(x + 1)/[x(x + 1)] + x/[x(x + 1)] = (2x + 1)/[x(x + 1)], x ≠ 0, −1
The subtraction trap. When subtracting, the minus sign applies to the whole second numerator, so bracket it before you distribute:
(3x + 1)/(x − 2) − (x − 4)/(x − 2) = [(3x + 1) − (x − 4)]/(x − 2) = (2x + 5)/(x − 2)
Without the brackets you would get 2x − 3, and the mark is gone.
Rational equations
The idea: Multiply every term by the LCD to clear the fractions, solve what is left, and then throw out any answer that was a non-permissible value all along.
Worked example: x/(x − 4) = 4/(x − 4) + 3.
- Restriction first: x ≠ 4.
- Multiply every term by (x − 4): x = 4 + 3(x − 4).
- Solve: x = 3x − 8, so −2x = −8 and x = 4.
- But x = 4 is the restriction, so it must be rejected. There is no solution.
Finding the restriction in step one is what makes step four obvious instead of surprising.
Work problems. These are rational equations in disguise, and the trick is to work in rates rather than times. If Amira paints a room in 3 hours and Ben in 6, then in one hour they do 1/3 + 1/6 = 1/2 of the room, so together they take 2 hours. Averaging 3 and 6 to get 4.5 is wrong on its face: together has to be faster than either alone.
What costs marks
The idea: Three habits fix nearly all of it.
- Write the restrictions first, from the factored denominators, before you touch anything else.
- Cancel factors, never terms. In (x + 3)/(x + 5) nothing cancels — those are terms, not factors. Only a factor that multiplies the whole numerator and the whole denominator can go.
- Bracket the second numerator when subtracting. Every single time.