Rational Expressions & Equations

Fractions with letters in them obey the fraction rules you already know — plus one forbidden value.

One tap fills the tub in 3 hours, the other in 6.

Turn both on. A tempting answer is the average, four and a half hours, but that cannot be right.

one tap fills it in 3 hours, the other in 6 3 h 6 h together? not 4.5 hours — two taps beat either one alone

Two taps must be faster than either one alone, so the answer has to be under 3 hours.

What adds is not the times, it is the rates: 13 of a tub per hour plus 16 gives 12.

Half a tub per hour means two hours. Adding rates means adding fractions with letters in them — which is this whole unit.

Where a rational function cannot go x = 2 y = 1 one input is forbidden; the far ends settle on one output

Simplify by factoring, never by crossing out terms

A rational expression simplifies only when top and bottom share a whole factor.

x2-9x+3 factors to (x-3)(x+3)x+3, and the (x+3) cancels to leave x - 3.

But x = -3 is still excluded, because the original expression was undefined there.

Factors cancel, terms do not

x+2x is not 2. The x on top is added, not multiplied, so nothing cancels. Factor first, then look.

Multiplying and dividing

Multiplying is the easy one: factor everything, cancel across, and multiply what is left.

Dividing is multiplying by the reciprocal, exactly as with numbers. Flip the second fraction and carry on.

Cancel before you multiply out — expanding first creates large polynomials you then have to factor again.

Cancel early

Factoring first turns a page of algebra into two lines. The temptation to expand is almost always the longer road.

Adding needs a common denominator

As with numbers, you cannot add until the bottoms match. Factor each denominator to find the least common denominator.

For 1x + 1x+2 the LCD is x(x+2), so the fractions become x+2x(x+2) and xx(x+2).

Add the tops, leave the bottom, then factor the result to see whether anything cancels.

Multiply the tops too

Rewriting 1x over x(x+2) means multiplying top and bottom by (x+2). Changing only the denominator changes the value.

Solving: clear the fractions

An equation is different from an expression — here you may multiply every term by the LCD and be rid of the fractions entirely.

That turns a rational equation into a polynomial one, which you already know how to solve.

Then check each answer against the excluded values, because multiplying by the LCD can introduce a root that was forbidden all along.

Expression or equation?

You may clear denominators only when there is an equals sign. In an expression, multiplying by the LCD changes what you were given.

Asymptotes

An excluded value that does not cancel gives a vertical asymptote — the graph shoots off beside that line.

For the far left and right, compare the degrees: equal degrees give a horizontal asymptote at the ratio of the leading coefficients.

If the bottom has the higher degree, the graph settles on y = 0, because the denominator wins the race.

A hole, not an asymptote

If every copy of the denominator factor cancels and the reduced expression is defined at that value, the graph has a hole. If a denominator factor remains, there is still a vertical asymptote.

Worked examples

Simplifying

Simplify x2 - 4x2 + 5x + 6 and state the excluded values.

  1. Factor both: (x-2)(x+2)(x+2)(x+3). Nothing can cancel until both are in factors.
  2. Cancel (x+2) to get x-2x+3. A whole factor, top and bottom.
  3. Excluded: x = -2 and x = -3. Both came from the original denominator, and -2 stays excluded even though it cancelled.

x-2x+3, x ≠ -2, -3

Adding

Simplify 3x + 2x+1.

  1. The LCD is x(x+1). The denominators share nothing, so the LCD is their product.
  2. Rewrite: 3(x+1)x(x+1) + 2xx(x+1). Multiply each top by whatever its bottom was missing.
  3. Add the tops: 3x + 3 + 2xx(x+1) = 5x+3x(x+1). The top does not factor, so this is finished.

5x+3x(x+1)

The two taps

One tap fills a tub in 3 hours, another in 6. How long together?

  1. Rates add: 13 + 16 of a tub per hour. Times do not add; rates do.
  2. Common denominator 6: 26 + 16 = 36 = 12. Half a tub per hour.
  3. Half a tub per hour means one tub in 2 hours. Under 3, as it had to be.

2 hours

Words to know

Rational expression
A fraction whose top and bottom are polynomials.
Excluded value
An input that makes the denominator zero. It can never be in the domain.
Vertical asymptote
A vertical line the graph approaches but never reaches, at an excluded value that does not cancel.
Horizontal asymptote
The height the graph settles towards far out to the left and right.
Least common denominator
The smallest expression every denominator divides into.

Mistakes to watch for

Cancelling terms instead of factors. In x+2x nothing cancels, because the x on top is added rather than multiplied. Only a factor shared by the whole top and the whole bottom may go.

Losing an excluded value after it cancels. x2-9x+3 simplifies to x-3, but x = -3 is still forbidden — the original expression was undefined there, and simplifying does not undo that.

Adding the denominators. 1x + 1x+2 is not 22x+2. Find a common denominator and add only the tops, exactly as with numbers.

Clearing denominators in an expression. Multiplying by the LCD is allowed only across an equals sign. In an expression it changes the value you were asked to simplify.

Averaging the times in a work problem. Two taps taking 3 and 6 hours do not average to 4.5; together they must beat 3. Add the rates, not the times.

Work problems like this appear in a Chinese text from around 100 BC, with the same tubs and the same trap.

The forbidden values, where the bottom is zero, are what give rational graphs their vertical asymptotes — the lines the curve races towards and never touches.

Engineers call the ratio of two polynomials a transfer function, and use it to describe everything from a loudspeaker to a suspension bridge.

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